This paper investigates the relationships and projection properties between vector-valued and scalar Jack and Macdonald polynomials, focusing on their algebraic symmetries and representation-theoretic aspects.
Contribution
It establishes conditions for projections that preserve symmetry actions and explores the connection between nonsymmetric and highest weight symmetric polynomials, including clustering conjectures.
Findings
01
Identified conditions for projections commuting with symmetric group and Hecke algebra actions.
02
Analyzed the relation between nonsymmetric and highest weight symmetric polynomials.
03
Studied the quasistaircase partition in the context of clustering conjectures.
Abstract
We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of the representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.
Equations204
inv(S):=#{(i,j):1≤i<j≤N,CTS[i]≥CTS[j]+2}.
inv(S):=#{(i,j):1≤i<j≤N,CTS[i]≥CTS[j]+2}.
α(S)i=v(rowS[i]), for 1≤i≤N and S∈Tabτ,
α(S)i=v(rowS[i]), for 1≤i≤N and S∈Tabτ,
\displaystyle\left\{\begin{array}[]{ll}(T_{i}+1)(T_{i}-t)=0,&\text{ for }1\leq i<N-1,\\[7.22743pt]
T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1},&\text{ for }1\leq i<N-2,\text{ and }\\[7.22743pt]
T_{i}T_{j}=T_{j}T_{i},&\text{ for }1\leq i<j-1\leq N-2.\end{array}\right.
\displaystyle\left\{\begin{array}[]{ll}(T_{i}+1)(T_{i}-t)=0,&\text{ for }1\leq i<N-1,\\[7.22743pt]
T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1},&\text{ for }1\leq i<N-2,\text{ and }\\[7.22743pt]
T_{i}T_{j}=T_{j}T_{i},&\text{ for }1\leq i<j-1\leq N-2.\end{array}\right.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Geometry · Advanced Topics in Algebra
Full text
Connections between Vector-valued and highest weight Jack and Macdonald polynomials
Laura Colmenarejo
Laura Colmenarejo, University of Massachusetts at Amherst, US
Jean-Gabriel Luque, Université de Rouen Normandie,
Laboratoire d’Informatique, du Traitement de l’Information et des Systèmes (LITIS), Avenue de l’Université - BP 8, 76801 Saint-Étienne-du-Rouvray Cedex, France
We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, specially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.
Key words and phrases:
Macdonald and Jack Polynomials, singular polynomials, highest weight polynomials, vector-valued polynomials, representation theory of symmetric group and Hecke algebra
This work is partially supported by the “European Regional Development Fund” (ERDF) via the regional (GRR) project MOUSTIC
A singular polynomial is a polynomial belonging to the kernel of all the Dunkl operators [13]. These operators are of prime importance in the study of polynomial representations of Hecke algebras and, in particular, in the study of Macdonald polynomials (see [27]). Macdonald polynomials are two parameters (q,t) multivariate polynomials indexed by compositions or partitions which can be symmetric or nonsymmetric and homogeneous or shifted, and which degenerate to (one parameter κ) Jack polynomials when setting q=tκ and sending t to 1. For certain values of the parameters (q,t) (respectively κ), some nonsymmetric Macdonald (respectively Jack) polynomials indexed by special compositions are singular. The singular Macdonald polynomials have important properties; for instance, singular shifted Macdonald polynomial equals the corresponding homogeneous one. The notion of singularity has a symmetric counterpart called highest weight. Macdonald polynomials are highest weight if they belong to the kernel of the sum of the Dunkl operators. For Jack polynomials, the highest weight polynomials are characterized similarly in terms of the partial sums.
We propose to investigate the relationship between the notions of singularity and highest weight by using a more general class of polynomials, called vector-valued Macdonald and Jack polynomials [14, 15], whose coefficients belong to a representation of the Hecke algebra and which project to the classical case while preserving singularity properties. This work is part of a larger study [9, 16, 20] on the conjectures of Bernevig and Haldane [3] on clustering properties of (symmetric) Jack polynomials.
These clustering properties are closely related to the quasistaircase partition, which get our attention later on the paper. Taking the Jack polynomials case as a guide, we replicate the study for Macdonald polynomials. Our study is based on Theorem 5.9, which is already included in [9] as a conjecture as part of our conclusions and that will be proved in a forthcoming paper of the authors, [8].
The paper is organized as follows. In Section 2, we set up the representation theory framework for the symmetric group, the Hecke algebra and the polynomials representations. This section also includes a very brief description of the combinatorial objects appearing in our study. In Section 3, we introduce the vector-valued polynomials together with their projections and the characterization of the symmetric elements in the general setting of polynomials. In Section 4, Jack and Macdonald polynomials are defined and we state some consequences related to the previous section. Section 5 is devoted to the quasistaircase partition. Starting with the importance of the quasistaircase and its definition, we analyze the singularity of nonsymmetric Jack polynomials and its consequences for the highest weight symmetric Jack polynomials. In parallel fashion, we finish this section with the analogous analysis for the Macdonald polynomials. In Section 6, we investigate some consequences of our results and we exhibit some factorization formulas. We finish this paper wrapping up our conclusions and perspective in Section 7.
2. Representation theory
Jack and Macdonald polynomials are constructed by use of the representation theory of the symmetric group and the Hecke algebra. In this section we refresh basic concepts and results with the aim of setting up the notation and the starting point.
2.1. Combinatorial objects
Let us start recalling the definition of combinatorial objects that appear in our study and setting our notation.
A partitionτ=(τ1,…,τN) is a nonincreasing sequence such that τi≥0, for all i. The length of a partition τ is the number of nonzero parts of τ, ℓ(τ)=max{i:τi>0}. Moreover, we say that τ is a partition of n, or that the size of τ is n, if ∑iτi=n. We denote by τ⊢n or ∣τ∣=n if τ is a partition of n and by Par(n) the set of partitions of n. We consider the following partial order on partitions: For τ,γ∈Par(n), we say that τ dominates γ, and we write τ≻γ, if τ=γ and i=1∑jτi≥i=1∑jγi, for all 1≤j≤n.
A compositionα=(α1,…,αN) is any permutation of a partition. We denote by α+ the unique nonincreasing rearrangement of α such that α+ is a partition and by ∣α∣ the size of the composition, that is, the sum of all its parts. We denote by Comp the set of compositions.
The definition of the partial order on partitions applies also for compositions since it does not use that the sequences are weakly decreasing. Moreover, it can be used to define another order: For α and β compositions, we write α▹β if ∣α∣=∣β∣ and either α+≻β+, or α+=β+ and α≻β.
Remark 2.1**.**
Notice that, by definition, the partitions and compositions appearing in this paper are allowed to have zeros and are standarized to have N entries in total (including the zeros). However, we omit the zero entries in those partitions for which they are not relevant. Moreover, we mostly work with partitions of N, Par(N). That is, the set of partitions τ=(τ1,…,τN) with ∑iτi=N.
A Ferrers diagram of shapeτ∈Par(n) is obtained by drawing boxes at points (i,j), for 1≤i≤ℓ(τ) and 1≤j≤τi (corresponding to French notation). Given a Ferrers diagram of shape τ, we define the conjugate partitionτ′ as the partition associated to the diagram that is obtained by exchanging rows and columns.
We define three fillings of a Ferrers diagram of shape τ∈Par(n):
•
A column-strict Young tableau is a filling such that the entries are increasing in the columns and nondecreasing in the rows, and they are in {1,2,…,n}.
•
A reverse standard Young tableau (RSYT) is a filling such that the entries are exactly {1,2,…,n} and are decreasing in rows and columns.
•
A reverse row-ordered standard tableau is a filling such that the entries are exactly {1,2,…,n} and are decreasing in rows, with no condition on the columns.
This paper has the reverse standard Young tableaux as one of the main combinatorial objects. Therefore, we denote by Tabτ the set of RSYT of shape τ and let Vτ be the space spanned by RSYTs of shape τ with orthogonal basis Tabτ. Note that Tabτ⊂RSTτ, where RSTτ denotes the set of reverse row-ordered standard tableaux of shape τ.
We finish this subsection introducing useful notation for the tableaux in Tabτ.
Let S∈Tabτ. The entry i of S is at coordinates (rowS[i],colS[i]) and the content of the entry is CTS[i]:=colS[i]−rowS[i]. Then, each S∈Tabτ is uniquely determined by its content vectorCTS=[CTS[i]]i=1N.
For instance,
S=\scalebox0.7\leavevmodeto57.31pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt14.22638pt\pgfsys@moveto28.45276pt14.22638pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto42.67914pt28.45276pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@closepath\pgfsys@moveto42.67914pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto42.67914pt0.0pt\pgfsys@moveto42.67914pt0.0pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto56.90552pt14.22638pt\pgfsys@lineto56.90552pt0.0pt\pgfsys@closepath\pgfsys@moveto56.90552pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke7\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke6\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.047.29233pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture has shape τ=(4,3) and content vector CTS=[1,3,0,−1,2,1,0].
There is a partial order on Tabτ related to the inversion number:
[TABLE]
We denote by S0 the inv-maximal element of Tabτ, which has the numbers N,N−1,…,1 entered column-by-column, and by S1 the inv-minimal element of Tabτ, which has these numbers entered row-by-row.
Given S∈Tabτ, and two nonnegative integers, m and m0, such that m0 is a factor of m, we associate to S the sequence α(S) defined by setting
[TABLE]
where v is the auxiliary function v: v(1)=0, and v(j)=m+(j−2)m0, for 2≤j≤ℓ(τ). The sequence α(S) has the property that in every final part of the sequence any number α(S)i occurs at least as often as the number α(S)i+1. This property resembles to the definition of reverse lattice permutations, also known as Yamanouchi words [27], and that is why we call α(S) the reverse lattice permutation of S.
For instance, for m=2, m0=1 and
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ke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt32.34373pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke8\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt32.34373pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt46.57011pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke7\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture, we have
α(S)=(0,3,0,2,0,0,4,3,2,0). We do not specify the integers m and m0 in the notation since it will be clear by the context.
2.2. The symmetric group and the Hecke algebra
Let SN be the symmetric group. That is, it is the set of permutations of {1,2,…,N}, which acts on CN by permutation of the coordinates. The group SN is generated by simple reflections si:=(i,i+1), for 1≤i<N, and it is abstractly presented by {si2=1:1≤i<N}, together with the braid relations (si+1sisi+1=sisi+1si and sisj=sjsi for ∣i−j∣>1). The associated group algebra CSN is the linear space {σ∈SN∑cσσ}.
The Jucys-Murphy elements for CSN are the elements of the form ωi:=j=i+1∑Nsij, for 1≤i<N, where sij denotes the transposition (i,j).
The symmetric group can be seen as a particular case of the Hecke algebra, HN(t), defined for a formal parameter t as the associative algebra generated by {T1,T2,…,TN−1} subject to the following relations:
[TABLE]
The Jucys-Murphy elements for HN(t) are defined recursively by ϕN=1 and ϕi=t1Tiϕi+1Ti, for 1≤i<N.
For t=1, CSN and HN(t) are identical. Otherwise, there is a linear isomorphism based on the map si⟼Ti.
2.3. Irreducible representations
We start this subsection recalling the definitions of a representation and an irreducible representation in the general framework of group theory.
A representation of a group G on a vector space V is a group homomorphism from G to GL(V), the general linear group on V. It is common practice to refer to V itself as the representation or G-module. A subspace W of V that is invariant under the group action is called a subrepresentation. If V has exactly two subrepresentations, namely the zero-dimensional subspace and V itself, then the representation is said to be irreducible.
We describe now the irreducible representations for each case. For the Hecke algebra, the irreducible representations are indexed by partitions of N and, by abuse of notation, we denote them by its indexing partition τ⊢N. The representations are constructed in terms of RSYT and the action of {Ti} on the basis elements. For S∈Tabτ and i, with 1≤i<N,
(I)
If rowS[i]=rowS[i+1], then Sτ(Ti)=tS.
2. (II)
If colS[i]=colS[i+1], then Sτ(Ti)=−S.
3. (III)
If rowS[i]<rowS[i+1] and colS[i]>colS[i+1], we denote by S(i)∈Tabτ the tableau obtained from S by exchanging i and i+1 and b=CTS[i]−CTS[i+1]. Then, Sτ(Ti)=S(i)+1−t−bt−1S.
4. (IV)
If CTS[i]−CTS[i+1]≤−2, then Sτ(Ti)=(tb−1)2t(tb+1−1)(tb−1−1)S(i)+tb−1tb(t−1)S, where we use the same notation than in the previous case.
Observe that the last case can be obtained from the previous case by interchanging S and S(i) and applying the quadratic relation τ(Ti)2=(t−1)τ(Ti)+tI, where I denotes the identity operator on Vτ. We will refer to the formulas (I–IV) as the action formulas for τ(Ti).
Taking t=1, we recover the irreducible representation for the symmetric group SN and the action of τ(si). We describe them here. For S∈Tabτ and i, with 1≤i<N, there are four possibilities:
(I)
If rowS[i]=rowS[i+1], then Sτ(si)=S.
2. (II)
If colS[i]=colS[i+1], then Sτ(si)=−S.
3. (III)
If rowS[i]<rowS[i+1] and colS[i]>colS[i+1], then Sτ(si)=S(i)+b1S, where b=CTS[i]−CTS[i+1] as in the Hecke algebra case.
4. (IV)
If CTS[i]−CTS[i+1]≤−2, then Sτ(si)=(1−b21)S(i)+b1S.
We will refer to these formulas as the action formulas for τ(si).
For instance, the Jucys-Murphy elements for CSN act on S∈Tabτ by Sτ(ωi)=CTS[i]S, whereas for HN(t), the Jucys-Murphy elements act on S∈Tabτ by Sτ(ϕi)=tCTS[i]S.
Consider the following inner product on Vτ: For S,S′∈Tabτ, ⟨S,S′⟩t:=δS,S′⋅γ(S,t), with
[TABLE]
This inner product degenerates to the inner product in SN:
[TABLE]
2.4. Polynomial representations
For N≥2, let us denote by x the set of variables {x1,x2,…,xN}. For a composition α∈Comp, let xα=∏ixiαi be a monomial of degree n=∣α∣. Let F be an extension field of C, possibly C(κ) or C(q,t), for κ, q and t transcendental or formal parameters. Consider the space of polynomials, P:=spanF{xα:α∈Comp} and the space of homogeneous polynomials Pn:=spanF{xα:α∈Comp,∣α∣=n}, for n∈Z>0.
The action of SN on polynomials is defined by saying that si permutes the variables xi and xi+1. Therefore, p(x)si:=p(xsi), for 1≤i<N. For arbitrary transpositions, sij exchanges the positions of xi and xj and p(x)sij:=p(xsij). In general, p(x)σ=p(xσ−1), where (xσ)i=xσ−1(i), since it is an action on the right.
The action of the Hecke algebra HN(t) on polynomials is defined by
[TABLE]
The defining relations can be verified straightforwardly.
The following concept will have a relevant role throughout the paper since it characterizes the space of polynomials that transforms according to a specific τ.
Definition 2.2**.**
Suppose V is a linear space of polynomials which is invariant under the actions of SN or HN(t). We say that the elements of V are of isotypeτ if there is a basis {gS:S∈Tabτ} which transforms under the action formulas for τ(si) or τ(Ti), respectively.
The following construction shows a canonical space of isotype τ for the action of HN(t) whose elements are of minimal degree. Setting t=1 provides the definition for the action of SN. First, we define the polynomial associated to S0 as
[TABLE]
Now, if rowS[i]<rowS[i+1] and colS[i]>colS[i+1], then
[TABLE]
respectively. Since inv(S(i))=inv(S)+1, hS is determined by {hS′:inv(S′)>inv(S)} and can be computed from hS0 applying successively (4).
For example, consider τ=(2,1), for which there are two RSYT of this shape,
S0=\scalebox0.7\leavevmodeto28.85pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture and S1=\scalebox0.7\leavevmodeto28.85pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture .
By (3), hS0(x)=tx2−x3. Moreover, by (4),
hS0(x)T1=(t−1)tx2+t2x1−tx3, and so hS1(x)=t2x1−t+1t(x2+x3).
Remark 2.3**.**
The dominant monomial in hS0 is the product of the dominant monomials of the Vandermonde determinants associated to the columns. For instance consider
S0=\scalebox0.7\leavevmodeto43.08pt\vboxto43.08pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt28.45276pt\pgfsys@moveto0.0pt28.45276pt\pgfsys@lineto0.0pt42.67914pt\pgfsys@lineto14.22638pt42.67914pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@closepath\pgfsys@moveto14.22638pt42.67914pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke6\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt32.34373pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture, the dominant monomial in hS0 is x42x5⋅x2⋅1.
2.5. More operators
In this subsection we present several fundamental families of operators acting on polynomials. The Jack and Macdonald polynomials are defined through them, but these operators are also an important tool for our study.
Before introducing the operators, it is important to point out that we work with operators acting on the right. It is less common, but it follows the reading direction. For instance, if we look at the partial derivatives, ∂i=∂xi∂, and the divided differences, ∂ij=(1−sij)xi−xj1 as right operators, then
[TABLE]
We start defining the operators for the symmetric group case. Let κ be a parameter.
The Dunkl operators are defined for 1≤i≤N as:
[TABLE]
and the Cherednik operators are defined as:
[TABLE]
The operators Di and Ui commute with each other. Moreover, the last ones satisfy the following commutation relations: siUisi=Ui+1+κsi, for all 1≤i<N, and siUj=Ujsi, for i<j or j+1<i (see [27]).
We also have the operator ω that acts as the product of all the simple transpositions ω=s1s2⋯sN−1.
For the Hecke algebra case, let q and t be parameters. Then, the (q,t)-Dunkl operators are defined recursively as
[TABLE]
and the (q,t)-Cherednik operators are defined as
[TABLE]
Since the only repeated notation appears in the Dunkl operators, we include the superscript q,t to avoid confusion with the group case.
Note that the (q,t)-Cherednik operators satisfy the recursion: ξi=t1Tiξi+1Ti, and that they commute with each other (see [1]).
2.6. Singularity and the properties SP and SMP
We introduce two definitions for each of our frameworks, and one of the goals of this paper is to relate them.
The first concept is a property that characterize some polynomials.
Definition 2.4**.**
A polynomial p with rational coefficients is called singular in the framework of the symmetric group if there is a rational number κ0 such that, for all i, pDi=0 for κ=κ0.
Analogously, a polynomial p is singular in the framework of the Hecke algebra if, for all i, pDi=0 for a specialization of the value of (q,t) (such as q=t−4).
Let us illustrate this with an example. Consider p(x)=1≤i<j≤3∏(txi−xj) and the specialization q=t−2. Then, p(x) is singular since a direct computation shows that pTi=−p, for i=1,2. In fact, pT1T2=p and p(x)ωq=(tqx3−x1)(tqx3−x2)(tx1−x2)=t−2p(x). Furtheremore, by specializing qt=1, we obtain directly that pξ3=p.
The second concept is related to the bases.
Definition 2.5**.**
We say that a basis {gS:S∈Tabτ} satisfies the property (SP) if it is a basis for a space of polynomials of isotype τ for SN and each gS is singular, all with κ=κ0. Analogously, we say that a basis {gS:S∈Tabτ} satisfies the property (SMP) if it is a basis for a space of polynomials of isotype τ for HN(t) and each gS is singular, all with the same specialization of (q,t). Our study in Sections 5 and 6 are focused on specializations of the form qmtn=1, with 2≤n≤N, and possibly subject to some other conditions, which are the most common specializations.
3. Vector-valued polynomials
We consider the space of vector-valued polynomials Pτ=C[x1,x2,…,xN]⊗C[Tabτ], for a given partition τ⊢N. Then, we describe Pτ=P⊗Vτ as the span of xα⊗S, for α∈Comp and S∈Tabτ.
3.1. Definitions and action of the operators
Each of the operators defined in Section 2 has its counterpart in the vector-valued polynomial space. The following table summarizes all the operators already defined, both in the symmetric group and Hecke algebra case, together with their vector-valued version.
[TABLE]
Note that we use bold font for the operators that act on vector-valued polynomials. This way, it will be clearer on which space we act and our results will be more nicely presented.
3.2. Projections
We now set up a projection map from vector-valued polynomials to scalar ones for each framework that also behave well with some of the operators.
Proposition 3.1**.**
Let {gS:S∈Tabτ} be a basis for a space of polynomials of isotype τ for SN. The linear map
[TABLE]
intertwines tσ with σ, for all σ∈SN; that is, tσρ=ρσ.
Proof.
It follows immediately from the definition of {gS:S∈Tabτ}. For instance, taking σ=si and applying the action formulas for τ(si), we have that
[TABLE]
∎
As before, the basis {gS:S∈Tabτ} provide us a linear map that, roughly speaking, commutes with Ti and we have the Hecke algebra version of the previous result.
Proposition 3.2**.**
Let {gS:S∈Tabτ} be a basis for a space of polynomials of isotype τ for
HN(t). Then, the linear map
[TABLE]
intertwines Ti with Ti, for 1≤i<N; that is, Tiρ=ρTi.
Proof.
This result follows from the study of [(fS(xsi)⊗Sτ(Ti)]ρ according to the action of τ(Ti) and the fact that
[TABLE]
∎
3.3. Characterization of the symmetric elements
In this section, we characterize the vector-valued polynomials of Pτ that are symmetric under the two actions, for a fixed partition τ of N.
We start with the action of the symmetric group.
Theorem 3.3**.**
Consider p=S∈Tabτ∑pS⊗S, with pS∈P for all S. Then, psi=p, for all 1≤i<N, if and only if, for each S∈Tabτ, pS=⟨S,S⟩11gS, where {gS:S∈Tabτ} is a basis for the polynomials of isotype τ.
Proof.
Assume that psi=p, for all 1≤i<N. This equation can be written, for a fixed i, as
[TABLE]
As usual in these arguments, we work with invariance for each simple reflection si, which suffices to prove the symmetry. For that, we analyze the action of si according to the action formulas for τ(si) over pS.
(I)
For rowS[i]=rowS[i+1], pS(xsi)⊗(Sτ(si))=pS(xsi)⊗S=pS(x)⊗S, thus pS(xsi)=pS(x).
2. (II)
For colS[i]=colS[i+1], pS(xsi)⊗(Sτ(si))=−pS(xsi)⊗S=−pS(x)⊗S, thus pS(xsi)=−pS(x).
3. (III)
For rowS[i]<rowS[i+1], S(i)∈Tabτ and the equation (8) requires that
[TABLE]
Using the action, we have that
[TABLE]
Matching up the coefficients and rewriting the equations, we obtain that
[TABLE]
which imply that
[TABLE]
Substitute pS=⟨S,S⟩11gS and pS(i)=⟨S(i),S(i)⟩11gS(i), then
[TABLE]
By definition, ⟨S,S⟩1⟨S(i),S(i)⟩1=1−b21, which shows that gS and gS(i) transform according to the relations.
4. (IV)
This last case follows from the previous one by interchanging S and S(i).
This finishes one implication.
To prove the converse, let {gS:S∈Tabτ} be a basis for the polynomials of isotype τ such that p=S∈Tabτ∑pS⊗S, where pS=⟨S,S⟩11gS, for each S∈Tabτ. Then, the previous study by cases proves that psi=p, for all 1≤i<N, if it is read backwards.
∎
Now, it is the turn for the Hecke algebra.
Theorem 3.4**.**
Consider p=S∈Tabτ∑pS⊗S, with pS∈P for all S. Then, pTi=tp,
for 1≤i<N, if and only if, for each S∈Tabτ, pS=⟨S,S⟩t1gS, where {gS:S∈Tabτ} is a basis for polynomials of isotype τ for HN(t−1).
Proof.
Assume that pTi=tp, for a fixed i.
We analyze the action of Ti according to the action formulas for τ(Ti) over pS.
(I)
For rowS[i]=rowS[i+1], we have
that pS(xsi)=pS(x) since
[TABLE]
2. (II)
For colS[i]=colS[i+1], we have that
[TABLE]
which is equivalent to
[TABLE]
That is pSTi(t−1)=−pS.
3. (III)
For rowS[i]<rowS[i+1], the idea is to introduce p∂i=xi−xi+1p(x)−p(xsi) and solve psi=p−(xi−xi+1)(p∂i). For doing this, we abbreviate f=pS and f′=pS(i) and we require that t(f⊗S+f′⊗S(i))=(f⊗S+f′⊗S(i))Ti. For the right side, we have that
[TABLE]
where the coefficients {a11,a21,a22} are given by the action formulas for τ(Ti). Replace fsi and f′si by the above expressions in f∂i and f′∂i. Matching up the coefficients of S and S(i) gives two equations which are solved for f∂i and f′∂i:
[TABLE]
Substitute these expressions in fTi(t−1) and f′Ti(t−1) to obtain
[TABLE]
Now, suppose that f=CTSgS and f′=CTS(i)gS(i), then {gS} satisfies the hypotheses for isotype τ in HN(t−1) provided CTSCTS(i)=γ(S(i);t)γ(S;t).
4. (IV)
This case follows from the previous one.
Note that this ratio is invariant under t→t−1, and one implication is proved.
For converse, let {gS:S∈Tabτ} be a basis for the polynomials of isotype τ such that p=S∈Tabτ∑pS⊗S, with pS=⟨S,S⟩t1gS, for each S∈Tabτ. Then, the previous study by cases proves that pTi=tp, for all 1≤i<N, if it is read backwards.
∎
3.4. Singularity and projections
We now explain the relation between the projections and the singular polynomials. For that, we first use the Jucys-Murphy elements to characterize the singular polynomials and then, we describe the relation in each framework. We start with the symmetric group case.
Lemma 3.5**.**
A polynomial g is singular for κ=κ0 if and only if
gUi=g+κ0gωi, where ωi are the Jucys-Murphy elements for SN.
Proof.
In general, it holds that (xig(x))Di=g(x)+xi(gDi)+κj=i∑g(xsij). Specializing to κ0, we have that gUi=g(x)+κ0j>i∑g(xsij). The converse is proved similarly.
∎
The following result sets up the key role of the singular polynomials in the projection map.
Proposition 3.6**.**
Suppose {gS:S∈Tabτ} satisfies the property(SP). Then, the map ρ defined in Proposition 3.1 intertwines Di, Ui with
Di, Ui respectively, for 1≤i≤N. That is, Diρ=ρDi and Uiρ=ρUi, for 1≤i≤N.
Proof.
For any f,g∈P and any κ, we have the general formula
[TABLE]
Let us specialize κ=κ0 and consider
[TABLE]
where gS,j corresponds to Sτ(sij).
Then, for each σ∈SN, there is a matrix [τ(σ)S,S′] such that Sτ(σ)=S′∈Tabτ∑S′τ(σ)S,S′. By definition gS(xσ)=S′∈Tabτ∑gS′(x)τ(σ)S,S′ and, in particular, gS,j(x)=S′∈Tabτ∑gS′(x)τ(sij)S,S′=gS(xsij). Thus,
[TABLE]
By hypothesis, gSDi=0 and hence, (fS⊗S)Diρ=(fS⊗S)ρDi. The fact that ρ commutes with each σ∈SN implies that (fS⊗S)Uiρ=(fS⊗S)ρUi, for each i.
∎
Recently, in [19], the authors study this projection via its interpretation as a homomorphism of standard modules of the rational Cherednik algebra.
Now, we present the analogous results for the Hecke algebra case.
Proposition 3.7**.**
A polynomial p is singular if and only if pξi=pϕi for 1≤i≤N, where ϕi are the Jucys-Murphy elements in HN(t).
Proof.
We proceed by induction.
By definition, pDN=0 if and only if pξN=p=pϕN.
Now, suppose that pDj=0, for i<j≤N, if
and only if pξj=pϕj, for i<j≤N. Then, 0=pDi=t1pTiDi+1Ti if and only if pTiDi+1=0, which in terms of the Jucys-Murphy elements, is equivalent to say that pTiξi+1=pTiϕi+1. Multiplying both sides on the right by Ti and applying the inductive hypothesis, we get that 0=pDi if and only if tpξi=tpϕi, and this completes the proof.
∎
The following result gives us an idea of how useful the property (SMP) can be.
Theorem 3.8**.**
Suppose {gS:S∈Tabτ} satisfies property (SMP). Then, the map ρ defined in Proposition 3.2 intertwines wq, Ei and Diq,t with ωq, ξi and Diq,t respectively, for 1≤i≤N. That is, wqρ=ρωq, Eiρ=ρξi and Diq,tρ=ρDiq,t, for 1≤i≤N.
Proof.
We analyze the commutation with wq since the other commutations follow from the definitions of ξi,Ei,Diq,t and Diq,t.
For wq, we have that
[TABLE]
where we use that t1−NgST1…TN−1=gSωq. To see this, consider the Jucys-Murphy element ϕ1. By Proposition 3.7, gSξ1=gSϕ1 and then,
[TABLE]
Cancelling out the factor TN−1…T1 gives the desired result.
∎
Remark 3.9**.**
Using a similar argument to Lemma 3.5, we show that
[TABLE]
Thus, if g is homogeneous of degree n, singular for κ=κ0 and of isotype τ, then these parameters satisfy that n+κ0(2N(N−1)−Σ(τ))=0, where Σ(τ) is the sum of the contents in the Ferrers diagram of τ (i.e. Σ(τ)=∑(i,j)∈τ(j−i)).
The role of this formula is more relevant for Macdonald polynomials indexed by quasistaircase partitions as we describe later in the subsection 5.4.3.
4. Jack and Macdonald polynomials
There are several types of Jack and Macdonald polynomials; for instance, symmetric and nonsymmetric, homogeneous and nonhomogeneous (so called shifted) etc. We are interested in the symmetric and nonsymmetric Jack and Macdonald polynomials, and in their vector-valued versions. In this section we present the ones that are relevant for our study. For more details see [15].
4.1. Jack polynomials
Consider a formal parameter κ. There is a basis of simultaneous eigenfunctions for the Cherednik operators, called nonsymmetric Jack polynomials{Jα(x)}, with ⊳-leading term xα for α∈Comp.
The eigenvectors ςα, called spectral vectors, are given by ςα(i)=αi+1+κ(N−r(α,i)), where r(α,i) is the rank function defined by
[TABLE]
Observe that the function i↦r(α,i) is a permutation of {1,2,…,N} and that α is a partition if and only if r(α,i)=i, for all i.
Using the commutation relations of si and Ui, it can be shown that
[TABLE]
The symmetric Jack polynomials, Jλ, are obtained by applying symmetrizing operators to the nonsymmetric Jack polynomial Jα such that α+=λ. In terms of operators, the symmetric Jack polynomials are the eigenvectors of ∑iUi. Moreover, for a partition λ and m≥1,
[TABLE]
To the question of how many of these sums suffice to separate different partitions, the answer is that considering m=1,2,…,N is certainly enough. Notice also that considering only m=1 does not work.
Finally, there are simultaneous eigenfunctions of {Ui}, called vector-valued nonsymmetric Jack polynomials, Jα,S, with (α,S)∈Comp×Tabτ. The eigenvectors are given by the spectral vectors ςα,S(i)=αi+1+κ⋅CTS[r(α,i)]. For more details, see [14].
4.2. Macdonald polynomials
Consider two formal parameters q and t. There is a basis of simultaneous eigenfunctions for the Cherednik-Dunkl operators, the nonsymmetric Macdonald polynomialsMα, labeled by α∈Comp, with ⊳-leading term t∗xα. For us, ∗ denotes the exponent of t, which is not relevant for the order on the monomials xα. In this case, the spectral vector is ζα(i)=qαitN−r(α,i). Notice that from the definition of the Cherednik-Dunkl operators, the Macdonald polynomials Mα are necessarily homogeneous.
As for Jack polynomials, Macdonald polynomials satisfy the following recursive formulas:
[TABLE]
The symmetric Macdonald polynomialsMλ are defined as the eigenvectors of ∑iξi and are indexed by partitions λ.
In this case, by the eigenfunction properties of symmetric Macdonal polynomials, for a partition λ and m≥1,
[TABLE]
As a remarkable difference between Jack and Macdonald polynomials, the sum for m=1 alone works for generic parameters q and t and the λi is determined by the exponent of q. This is an example of the meta-principle that adding a parameter can simplify the problem.
The vector-valued nonsymmetric Macdonald polynomials, Mα,S, are defined as the simultaneous eigenfunctions of the operators {Ei} in Pτ.
4.3. Projections and highest weight
As we mentioned before, the notion of singularity has a symmetric counterpart. Let us start with the definition and leave the motivation for the beginning of the next section.
Definition 4.1**.**
A polynomial is highest weight if it belongs to the kernel of the sum of the Dunkl operators ∑iDi, for the symmetric case, or to the kernel of ∑iDiq,t, for the Hecke algebra case.
The following two corollaries show that the projections defined in Section 3.2 in a more general setting restrict nicely to the Jack and Macdonald polynomials.
Corollary 4.2**.**
Under the same hypotheses as in Proposition 3.6, ρ maps each vector-valued nonsymmetric Jack polynomial to a scalar simultaneous {Ui}-eigenfunction, which is a Jack polynomial if the spectral vectors ζα for the corresponding degree determine α uniquely. Otherwise, it maps to zero.
Remark 4.3**.**
In general, there might be some negative rational κ values for which the spectral vectors ζα do not determine α uniquely.
Corollary 4.4**.**
Under the same hypotheses as in Proposition 3.8, ρ maps each vector-valued nonsymmetric Macdonald polynomial to a scalar simultaneous {ξi}-eigenfunction, which is a Macdonald polynomial if the spectral vectors ζα for the corresponding degree determine α uniquely. Otherwise, it maps to zero.
5. The quasistaircase partition
5.1. The importance of the quasistaircases
The investigation on highest weight Jack polynomial is motivated by the quest for the
description of the wave functions that model the fractional quantum Hall effect (FQH). The first occurrence
of a Jack polynomial (at that time we did not know it was one yet) is the Laughlin wavefunction itself [25]:
[TABLE]
This function depends on an integer parameter m and is known to be a true staircase Jack polynomial for the parameter α=−m1.
The study of φLm as a symmetric function generated literature for the purpose of the expansion in the Schur basis and the monomial basis [10, 33]. This first case is particularly interesting because it is also related to rectangular Jack polynomials and hyperdeterminants [2, 6, 26]. Notice that, even in this (simplest) case, Macdonald polynomial provides a more regular framework for the study of these functions. For instance, partitions involved in the expansion in the Schur basis are more easily understood for the q-deformation [5, 21].
Although the description of this wave function is not known in all the FQHE configurations, this particularly simple expression led physicists to note that it resulted from the coincidence of two notions: clustering properties and highest weight polynomials. The clustering properties was in particular studied by Feigin et al. in the early 2000s [17, 18] and described in terms of ideal, wheel conditions and staircase partitions. The second notion (highest weight) is related to some differential operators (or their q-deformations). A family of highest weight Macdonald polynomials was described in [20]. The part of the results concerning staircases can be recovered from the works of Feigin et al. while, for the polynomials indexed by quasistaircases which are not staircases, it is a non trivial consequence of the Lassalle binomial formulas [24].
Several Jack polynomials are used to describe some FQH states (or wave functions thought to be adiabatically related to the true eigenstates) [3, 4, 30, 32]. Nevertheless, it is not always the case and some FQH states seem not to be directly related to Jack polynomials. The hope of completing the picture with Macdonald polynomials comes from the theory of nonsymmetric Jack and Macdonald polynomials. These polynomials can be considered as elementary building blocks allowing to reconstitute symmetric polynomials by linear combinations. There is a double challenge with this approach. Firstly, we have to reconstruct the clustering and the highest weight notions from properties of nonsymmetric polynomials. Secondly, we have to write FQH wave functions nicely in terms of these elementary building blocks. In the present paper we deal with the first problem. Let us be more precise on this point.
The clustering properties comes from physical interpretation of constraints on the position of the particles. These constraints translate in terms of polynomials as vanishing properties. More precisely, the polynomial representing the wave function vanishes when k particles tends to the same position. The number k is one of the parameters describing the clustering property. A second parameters take into account the number of clusters which are required to observe the vanishing property and another parameter allows to take into account the strength with which particles repel each other. In terms of polynomial, the more strongly the particles repel each other, the faster the polynomial tends to [math] when k variables tend to the same value. For a mathematical purpose, instead of stating the
results in terms of clustering properties, we prefer an equivalent statement in terms of factorizations [9].
The computation of the exact wavefunctions being a very difficult problem, physicists overcome this difficulty by empirically research for polynomials that are adiabatically related to true eigenvalues and are guided by the clustering properties.
In this quest of wavefunctions, Bernevig and Haldane [3, 4], based on results from Feigin et al [17], noted that some of Jack’s polynomials indexed by quasistaircase partitions could be good candidates. In that context they gave three conjectures that we translate below in terms of factorization. We assume that the parameters k+1 and s−1 are coprime and we set λk,r,sβ=[(βr+s(r−1)+1)k,…,(s(r−1)+1)k,0n0] with n0=(k+1)s−1 and the number of particles (i.e. the size of the partition) is N=βk+n0. Notice that β is not really a parameter since it depends on N and the three other parameters. The three conjectures read:
(1)
First clustering property They considered s−1 clusters of k+1 particles
Z1=z1=⋯=zk+1, Z2=zk+1=⋯=z2(k+1),…,Zs−1=z(s−2)(k+1)+1=⋯=z(s−1)(k+1), together with
k particle cluster ZF=z(s−1)(k+1)=⋯=zs(k+1)−1.
The other particles (variables) remain free. For such a specialization,
the Jack polynomial Jλk,r,sβ−r−1k+1((k+1)(Z1+⋯+Zs−1)+kZF+zs(k+1)+⋯+zN) behaves as i=s(k+1)N∏(ZF−zi)r when
each zi (i=s(k+1),…,N) tends to ZF but has a remaining polynomial factor which does not tends to [math].
For instance,
J53(−2)(2Z1+ZF+z3+z4)=(ZF−z4)2(ZF−z3)2P(Z1,ZF,z3,z4) where P is a degree 4 polynomial.
2. (2)
Second clustering property They considered a cluster of n0=(k+1)s−1 particles z1=⋯=z(k+1)s−1=Z.
The Jack polynomial Jλk,r,sβ(−r−1k+1)(n0Z+zn0+1+⋯+zN) behaves as
i=s(k+1)∏N(Z−zi)(r−1)s+1 when each zi tends to Z. More specifically, for highest weight Jack polynomials, one has the
following explicit formula:
[TABLE]
3. (3)
Third clustering property It is obtained by forming s−1 clusters of 2k+1 particles Z1=z1=⋯=z2k+1,…,
Zs−1=z(s−2)(2k+1)+1=⋯=z(s−1)(2k+1). A highest weight Jack Jλk,r,sβ satisfies
[TABLE]
Notice that authors proved the second conjecture [9] and, although the proof is certainly more technical, we guess that the third conjecture should be proved following a similar method. The first conjecture seems more difficult because it involves a factor that has not been identified. Our strategy has been to extend these results in three directions :
(1)
To nonsymmetric polynomials in order to use the inductions involved in the Yang-Baxter graph;
2. (2)
To nonhomogeneous polynomials, because the vanishing properties are simpler to study in that context;
3. (3)
And to Macdonald polynomials in order to polarize the vanishing properties in the sense that, in the factorization formulas, the linear factors are pairwise distinct; this is easier to check algebraically.
Nonsymmetric Jack polynomials were introduced by Opdam [31] and were used to describe the polynomial part of the eigenfunctions of the Calogero–Sutherland model on a circle and define families of orthogonal polynomials which are multivariate analogues of Hermite and Laguerre polynomials [1]. Nonsymmetric Macdonald polynomials were introduced in [7, 28]. Marshall [29] tied the theories of symmetric and nonsymmetric Macdonald polynomials by adapting the approach of [1] to the (q,t)-deformation.
The nonsymmetric counterpart of the highest weight is the notion of singularity. As we define previously, a polynomial is singular if it belongs to the kernels of the Dunkl operators [12]. In [13] one of the authors shows that quasistaircase Jack polynomials, under some conditions, are singular. The notion of isotype for Jack polynomials used in the current paper comes from this article. One of our goals in this paper is to adapt the results and proofs developed in [13] to the Macdonald case.
Clustering properties of highest weight Macdonald polynomials are consequences of their factorizations into linear factors when submitted to some specializations. The main tool to show the factorization is another kind of Macdonald polynomials: the shifted Macdonald polynomials are the nonhomogeneous version of the symmetric Macdonald polynomials and have the additional property of being defined alternatively by vanishing conditions (see e.g. [22, 23]). The factorizations are obtained by combining homogeneity of highest weight shifted Macdonald polynomials together with their vanishing properties. These factorizations were investigated by the authors in two previous papers: for rectangular partitions in [16] and for general quasistaircases in [9].
Numerical evidences suggest that the clustering properties have nonsymmetric analogues. We gave several examples at the end of [9] that involve quasistaircase partitions. The
purpose of the present paper is to establish links between the notion of highest weight polynomials and singular polynomials. The problem of proving clustering properties of quasistaircase nonsymmetric Macdonald polynomials will be done in a future work.
5.2. The quasistaircase partition
The most general quasistaircase partition, or simply quasistaircase, is defined as follows.
Definition 5.1**.**
A quasistaircase partition is a partition of the form
[TABLE]
with 1≤ν≤n−1 and where al means that the part a occurs l times.
For instance, qs(5,3,2,4,7,2)=[172,123,73,23,07].
Instead of staying with this general definition, we look at a specific description of the quasistaircase. Consider a pair (m,n)∈N02, with 2≤n≤N
and nm∈/N0. Moreover, let d=gcd(m,n), m0=dm, n0=dn, κ0=−n0m0=−nm, and
l=⌈n0−1N−n+1⌉+1. We consider a partition τ=[n−1,(n0−1)l−2,τl] of N with τl≤n0−2. Then, we consider quasistaircase partitions of the form μ=qs(m0,n0−1,m,l−1,n−1,τl).
For instance, consider N=10, n=6, m=4, and τ=(5,2,2,1). Then, n0=3, m0=2, κ0=3−2 and μ=[8,6,6,4,4,05].
Although we have introduced more parameters in our description of the quasistaircase, they are advantageous for the specialization of the parameters (q,t) and the study of the Macdonald and Jack polynomials. Furthermore, from now on, μ denotes the quasistaircase partition μ=qs(m0,n0−1,m,l−1,n−1,τl).
5.3. The symmetric group case
In this subsection we summarize the work already done for the symmetric group case in order to provide the results that we use as a guide for the Hecke algebra case. We also give a brief overview of those proofs that are relevant.
5.3.1. Singularity of quasistaircase nonsymmetric Jack polynomials
The following proposition summarizes the results obtained in [13] related to the symmetric group case.
There exists an irreducible SN-module of isotype τ with Ferrers diagram μ=qs(m0,n0−1,m,l−1,n−1,τl), and constituted with polynomials that are singular for κ=κ0.
Moreover, for each S∈Tabτ, α(S)+=μ and the set {gS=Jα(S):S∈Tabτ} satisfies property (SP), where the nonsymmetric Jack polynomials are specialized to κ=κ0.
The following result describes the relation between S∈Tabτ, α(S), and κ0.
Proposition 5.3**.**
Consider S∈Tabτ and k such that 1≤k≤N. Then,
[TABLE]
Proof.
Fix k and S∈Tabτ, and let rowS[k]=i and colS[k]=j. Every entry u in the ith row with higher column index (to the right) has α(S)u=α(S)k with u<k. Also, every entry u in the pth row, with p>i, has α(S)u>α(S)k. Thus, r(α(S),k)=τi−j+1+p=i+1∑ℓ(τ)τp, and N−r(α(S),k)=j−1−p=1∑i−1τp.
We analyze α(S)k depending on i.
If i=1, then α(S)k=0 and N−r(α(S),k)=j−1=CTS[k].
Suppose now that τ has two rows and i=2, then N−r(α(S),k)=n−1+j−1 and α(S)k=m. Therefore,
[TABLE]
Finally, suppose τ has 3 or more rows and i≥2. By construction, α(S)k=m+(i−2)m0. Moreover, N−r(α(S),k)=j−1+(n−1)+(i−2)(n0−1). Then,
[TABLE]
∎
Note also that if some Jα is singular, then the SN-invariant subspace generated by the set {Jασ:σ∈SN} consists of singular polynomials, because sijDisij=Dj, for all i=j.
The following statements apply to the singular Jack polynomials Jα(S).
The polynomials Jα(S) with κ=−nm satisfy property (SP).
Proof.
The first step is to prove that Jμ is of isotype τ by means of a representation-theoretic argument about inducing the trivial representation of Sμ (the stabilizer of μ) up to SN [13, Theorem 5.2, Proposition 5.3][27]. This implies that any polynomial derived from Jμ by group action is of the same isotype.
It is clear that if rowS[i]=rowS[i+1], then α(S)i=α(S)i+1 and Jαsi=Jα. Suppose α(S)i<α(S)i+1, which is equivalent to rowS[i]<rowS[i+1]. Then, S(i)∈Tabτ, and
[TABLE]
The spectral vector satisfies ζα(S)(i)=α(S)i+1+κ0(N−r(α(S),i))=1+κ0CTS[i], thus
[TABLE]
Notice that α(S(i))=α(S)si by definition. It remains to show that Jα(S)si=−Jα(S) when colS[i]=colS[i+1]. By the action of the Jucys-Murphy elements, Jα(S)ωi=CTS[i]Jα(S). Consider the polynomial f=Jα(S)+Jα(S)si and the relation siωisi=ωi+1+si; then
[TABLE]
and similarly fωi+1=CTS[i]f. Also fωj=CTS[j]f, for j<i or j>i+1 (by the commutation siωj=ωjsi). Then f is of isotype τ but has impossible eigenvalues for each ωj, thus f=0.
∎
Remark 5.5**.**
Historically, the method of constructing singular polynomials was to show directly that Jμ is singular for certain partitions λ and then to analyze the SN-invariant subspace generated by Jμ. It turned out that this subspace corresponds to an irreducible representation of τ and this led to the correspondence between Jα and RSYTs of shape τ, where α is a reverse lattice permutation of μ.
Part of the proof of this fact concerns the property Jα(S)si=−Jα(S) when colS[i]=colS[i+1], and we sketch it in the proof of Theorem 5.4. More details of the argument establishing the τ-isotype property can be found in the proof of Theorem 5.10. In recent work, currently a paper under preparation, there is a direct proof of the isotype property which is used to prove singularity by means of the Jucys-Murphy elements in Lemma 3.5 and in Proposition 3.7.
Next we look at the implications for ρ. We describe sets of Jack polynomials {Jβ,S} which span irreducible SN-invariant subspaces of Pτ. We need another definition first.
Definition 5.6**.**
For α∈Comp and S∈Tabτ define ⌊α,S⌋ to be the filling of the Ferrers diagram of τ obtained by replacing i by αi+ in S, for all i.
This does not define a one to one correspondence, and so we consider the following set T(α,S)={(β,S′):⌊β,S′⌋=⌊α,S⌋}.
Jα,S* can be transformed to Jβ,S′ by a sequence of maps of the form f→af+bfsi if and only if ⌊α,S⌋=⌊β,S′⌋.
Furthermore, it is known that in this case, the spectral vectors of (α,S) and (β,S′) are permutations of each other.*
Next result identifies when there is a unique nonzero symmetric polynomial in terms of ⌊α,S⌋.
For (α,S)∈N0N×Tabτ, the span{Jβ,S′:(β,S′)∈T(α,S)} contains a unique nonzero symmetric polynomial if and only if ⌊α,S⌋ is a column-strict tableau.
For each shape τ, consider the unique minimal column-strict tableau of shape τ obtained by filling the ith row with the entries i−1 for each i. Then, the partition λ=((l−1)τl,(l−2)τl−1,…,1τ2,0τ1) and the inv-minimal RSYT S1 is the unique S such that ⌊λ,S⌋ equals this tableau.
For example, take τ=(5,2,2,1). Then, λ=(3,2,2,1,1,05),
[TABLE]
5.3.2. From singular nonsymmetric Jack polynomials to highest weight symmetric Jack polynomials
The fact that {Jα,S′} is a basis for Pτ and that there is a unique symmetric Jack polynomial of degree ∑i(i−1)τi implies that any symmetric polynomial of that degree is a scalar multiple of it. This minimal symmetric polynomial is constructed as follows. Consider the set of minimal polynomials of isotype τ, {hS:S∈Tabτ}, described in (3). Then, by the remark above, the polynomial Fτ:=S∈Tabτ∑⟨S,S⟩01hS⊗S is the symmetric Jack polynomial of minimal degree. Since i=1∑NFτDi is symmetric of lower degree, the sum vanishes. By [14, Thm. 5.22], Fτ=α+=λ∑cαJα,S1, with λ=[(l−1)τl,(l−2)n0+1,…,1n0+1,0(n−1]. One of the terms in Fτ is a nonzero multiple of xλ⊗S1 and there is no term of the form xλ⊗S′ with S′=S1 because the other terms in the Jα,S1 are of the form xβ⊗S′ with λ⊳β. This implies that one of the terms of hS1 is indeed xλ.
Now, we specialize to the hypotheses of Theorem 5.4 and apply Proposition 3.6. Let τ be a partition associated with a set of singular polynomials for κ=κ0. By (1) the leading term of Jα(S1) is xμ, where μ=qs(m0,n0−1,m,l−1,n−1,τl).
This implies that Fτρ=S∈Tabτ∑⟨S,S⟩01hSJα(S) has leading term xμ+λ (with a nonzero coefficient). Furthermore, ∑iFτρDi=0, for κ=κ0. Thus, Fτρ is a highest weight symmetric Jack polynomial [20].
Note that μ+λ=qs(m0+1,n0−1,m+1,l−1,n−1,τl).
5.4. The Hecke algebra case
We proceed to analyze the Hecke algebra case. To do so, for the rest of the discussion we assume the following result, whose proof is included in a forthcoming paper [8].
Theorem 5.9**.**
Let m=dm0 and n=dn0, for some d≥1, and g=gcd(m0,n0). Then, the polynomial
Mqs(m0,n0−1,m,l−1,n−1,τl) is singular for (q,t)=(zu−gn0,ugm0), where zgm0 is a gth root of unity. Equivalently, z=em02πik, where gcd(k,g)=1.
This theorem shows a very natural phenomenon. Hypothesize that there exist singular polynomials with properties analogous to the group case above. For m, n and d as in Theorem 5.9, consider
[TABLE]
Then, the theorem states that Mμ is singular for certain (q,t) satisfying qmtn=1.
As an example, take N=6, m=m0=3, n=n0=3 and τ=(2,2,2). Then, consider the quasistaircase μ=qs(3,2,3,2,2,2)=(6,6,3,3,0,0). The Macdonald polynomial Mμ is singular for q=zu−1,t=u, with z3=1,z=1. Another example is the quasistaircase μ=(243,163,011), for which the Macdonald polynomial Mμ is singular for the specializations q8t4=1 and q2t=±i.
5.4.1. Singularity of quasistaircase nonsymmetric Macdonald polynomials
In general, setting q=tκ1 and letting t→1 in Mα produces the Jack polynomial Jα. This should hold for singular polynomials, but not when we specialize to q=zu−n1 and t=um1, with z=1.
For instance, take λ=(2,0). Then, Mλ is singular for qt=−1, q2t2=1 but not for qt=1. Looking at the first specialization, qt=−1, the Macdonald polynomials is Mλ∣qt=−1=−(tx1−x2)(x1+x2) and there is no corresponding Jack polynomial.
This property indicates that the structure (labeling, isotype, etc.) of singular Macdonald polynomials is fairly close to that of the singular Jack polynomials, but still different enough to be interesting to study.
Theorem 5.10**.**
Consider the singular polynomial Mμ for the specialization qmtn=1 and the quasistaircase μ described in Theorem 5.9 (and possibly also a more restrictive condition). Then, Mμ is of isotype τ and {Mα(S):S∈Tabτ}, where α(S) is the reverse lattice permutation associated to S, is a basis for polynomials of isotype τ for HN(t).
Proof.
We start computing the spectral vector of α(S):
[TABLE]
It is clear that if rowS[i]=rowS[i+1], then α(S)i=α(S)i+1 and Mα(S)si=Mα(S).
In particular MμTi=tMμ, whenever μsi=μ. That is, si is a generator of the stabilizer group of μ, isomorphic to the product Sτ1×Sτ2×⋯×Sτl. Thus Mμ is in the HN(t)-module induced up from the trivial representation of Hτ1(t)×Hτ2(t)×⋯×Hτl(t). By the results of [11], this is a direct sum of components labeled by partitions γ each of which satisfies γ⪰τ. Furthermore, γ∈Par(N) and γ≻τ, which implies (i,j)∈γ∑(j−i)>(i,j)∈τ∑(j−i). By a result of Macdonald [27, (1.15)], it suffices to prove this for γ, γ′ such that γk=γk′+1 and γm=γm′−1, for some k<m (a raising operator). Then,
[TABLE]
The operator i=1∏Nϕi has the eigenvalue tp, for p=(i,j)∈γ′∑(j−i)=∑iCTS1, on polynomials of isotype γ. Since Mαi=1∏Nϕi=tpMα, and using the above inequality for eigenvalues, we see that Mα is of isotype τ.
Suppose α(S)i<α(S)i+1. We know that rowS[i]<rowS[i+1] and S(i)∈Tabτ. Then,
[TABLE]
Therefore, Mα(S) and S transform the same way under Ti.
By a similar argument as in Proposition 5.4, suppose colS[i]=colS[i+1]. Then, t1+CTS[i]=tCTS[i+1] and
[TABLE]
This implies that
[TABLE]
Similarly (Mα(S)+Mα(S)Ti)ϕi+1=tCTS[i](Mα(S)+Mα(S)Ti) and, together with Mα(S)Ti being of isotype τ, implies Mα(S)+Mα(S)Ti=0.
∎
Corollary 5.11**.**
The polynomials obtainable from Mμ by a sequence of maps f→af+bfTi are also singular.
Proof.
This is a consequence of the fact that fDi=0, for all i, implies fTjDi=0, for all i,j, which follows from equations (3.8) and (3.9) in [15].
∎
5.4.2. From singular nonsymmetric Macdonald polynomials to highest weight symmetric Macdonald polynomials
In [15] this case is already studied and we list a few useful results that are proved there.
For (α,S)∈Comp×Tabτ the span{Mβ,S′:(β,S′)∈T(α,S)} contains a unique nonzero symmetric polynomial if and only if ⌊α,S⌋ is a column-strict tableau.
As in the group case there is a unique symmetric polynomial of minimal degree. In this case, ∑iEi determines ⌊λ,S⌋ and the set T(λ,S). Indeed Mλ,S∑iEi=∑iqλitCTS[i]Mλ,S. The tableau ⌊λ,S⌋ can be constructed from the eigenvalues.
Since the entries 1, 2 and 3 in S can be arbitrarily permuted, there are 6 different RSYTs leading to the same ⌊γ,S⌋. Moreover, the diagram ⌊γ,S⌋ is column-strict, and therefore there is a symmetric polynomial in the SN-span. Finally, for the shape of S, (3,2,1), the corresponding minimal type is λ=(2,1,1,0,0,0), and S1=\scalebox0.7\leavevmodeto43.08pt\vboxto43.08pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt28.45276pt\pgfsys@moveto0.0pt28.45276pt\pgfsys@lineto0.0pt42.67914pt\pgfsys@lineto14.22638pt42.67914pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@closepath\pgfsys@moveto14.22638pt42.67914pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke6\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt32.34373pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture and
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Let {hS:S∈Tabτ} be the set of minimal polynomials of isotype τ for HN(1/t) described in (3). By Theorem 3.4, Fτ:=S∈Tabτ∑γ(S;t)1hS⊗S is the symmetric polynomial and eigenfunction of ∑iEi of minimal degree. By [15, Thm. 5.27], we can write Fτ=α+=λ∑cαMα,S1. One of the terms in Fτ is (a nonzero multiple of) xλ⊗S1 and there is no term of the form
xλ⊗S′ with S′=S1 because the other terms in the Mα,S1 are of the form xβ⊗S′ with λ⊳β. This implies that one of the terms of hS1 is indeed xλ.
Now, we specialize to the hypotheses of Theorem 5.10 and apply Theorem 3.8. Let τ be a partition associated with a set of singular polynomials for (q,t). By the construction in formula (1) the leading term of Mα(S1) is xμ.
This implies that Fτρ=S∈Tabτ∑γ(S;t)1hSMα(S) has the leading term xμ+λ (with a nonzero coefficient), where the partition μ+λ is the quasistaircase μ+λ=qs(m0+1,n0−1,m+1,l−1,n−1,τl).
It is known that Ti and Dj commute for j=i,i+1. Therefore, j=1∑i−1Dj+j=i+2∑NDj commutes with Ti. Using that tDi=TiDi+1Ti, we deduce the other two cases:
[TABLE]
Suppose that fsi=f, so fTi=tf. Then, apply the operators in the equation to f to get that fDiTi+fDi+1Ti=tf(Di+Di+1). The commutations imply that if f is symmetric, then ∑jfDjTi=t∑jfDj. The same computation can be done for the vector-valued operators Ti and Di, since they satisfy the same relations. This proves the following result.
Lemma 5.15**.**
If f in Pτ or in P is symmetric, then so is ∑ifDi or ∑ifDi, respectively.
As in the group case, we see that Fτ∑iDi=0 because Fτ is the minimal degree symmetric polynomial in Pτ.
Theorem 5.16**.**
The polynomial Fτ is proportional to the symmetric Macdonald polynomial Mμ+λ and is annihilated by ∑iDi. Hence, it is a highest weight Macdonald polynomial.
In particular, we have recovered that the symmetric Macdonald polynomial Mμ+λ is highest weight [20, Theorem II] from the singularity of the nonsymmetric Macdonald polynomial Mμ.
This shows that the study of singular nonsymmetric Macdonald polynomials is relevant for the understanding of the Bernevig and Haldane conjectures [3].
5.4.3. A note on specializations
From Proposition 3.7, a polynomial p is singular if and only if pξi=pϕi, for 1≤i≤N.
Now, suppose Mα is singular for some specific (q0,t0). In this case, instead of considering the sum of ξ as in the symmetric case, we consider the product. Since ∑i(N−r(α,i))=N2−((1+2+⋯+N)=21N(N−1), we have that
[TABLE]
Suppose further that Mα (specialized at (q0,t0)) is of isotype τ, for some partition τ of N. Then, the eigenvalue of i=1∏Nϕi acting on an HN(t)-module of isotype τ is tΣ(τ), where
[TABLE]
Thus (q0,t0) must satisfy the equation q∣α∣tN(N−1)/2−Σ(τ)=1.
In particular, we look at the quasistaircase partition qs(m,n−1,dm,dn−1,l−1,τl) and the isotype τ=(dn−1,(n−1)l−2,τl). Then,
[TABLE]
and
[TABLE]
Denoting A=n1(21N(N−1)−Σ(τ)) and after some computations, we get that (q,t) must satisfy qmAtnA=1. Note that this is only a necessary condition.
6. Factorizations at special points
In this section we examine the polynomials hS of minimal degree associated to S∈Tabτ, defined by formulas (3) and (4) in Section 2.4, when evaluated at special points. Typically these involve a smaller than N number of free variables. There are also factorizations of highest-weight symmetric Jack and Macdonald polynomials for certain parameter values. The two structures are tied together by the projection formulas of Sections 4.3 and 5. The key fact is that Macdonald polynomials of isotype τ, denoted by gS with S∈Tabτ and S=S0, can be shown to vanish at the special points. This also applies to polynomials hS, with S∈Tabτ of isotype τ with minimum degree. The proofs are worked out in detail for the Hecke algebra case and a limiting argument (t⟶1) is used to derive the symmetric group version.
6.1. The symmetric group case
We start with the definition of a family of relevant polynomials.
Definition 6.1**.**
For each S∈Tabτ, define the alternating polynomial by setting
[TABLE]
where τ′ denotes the transpose of the partition τ and S(i,j) denotes the entry in the ith row and the jth column.
We state the relevant results on evaluations at special points and factorization for the symmetric group case. These are simple consequences of analogous facts for the Hecke algebra. Therefore, references to their proofs are given later on this paper, Remarks 6.15 and 6.17. We start describing the specialization.
Definition 6.2**.**
Let l:=ℓ(τ) and for 1≤i≤l−1, define the sequence given by
[TABLE]
Moreover, for notational convenience, we set nl:=N+1.
For the free variables z1,…,zτl, y1,…yl−1, define the special pointx∈RN by
[TABLE]
Our first remark is that with the sequence {ni}, the rows of S1 are, from bottom to top, [N,…,nl−1], [nl−1−1,…,nl−2], …, [n2−1,…,n1], [n1−1,…,1]. Our second remark is that we visualize the special point in the following way. For a shape τ, fill the top row with the variables zi from right to left. Then, for 1≤j≤ℓ(τ)−1, all the entries of the jth row are filled with yj.
In this way, the special point x corresponds to the reading of the tableau from top to bottom and from left to right.
For instance, for τ=(3,3,2),
[TABLE]
and the special point is x=(z1,z2,y2,y2,y2,y1,y1,y1).
Now, we present the two results that describe the polynomials hS at the special point (x), for all S∈Tabτ. As mentioned before, the proofs are consequences of the Hecke algebra case, which we investigate in Section 6.2.
Theorem 6.3**.**
Let S∈Tabτ be such that S=S1. Then, hS(x)=0.
Corollary 6.4**.**
hS1(x)=aS1(x).
Recall the projection of the symmetric Jack polynomial of minimal degree described in Section 5.3.2:
[TABLE]
where the leading term of Jα(S1) is xμ with
[TABLE]
This polynomial is singular for κ=−nm∈/N, with 2≤n≤N, m0=gcd(m,n)m≥2 and n0=gcd(m,n)n≥2. Moreover, it is of isotype τ=(n−1,(n0−1)l−2,τl), so that 1≤τl=N−(n−1)−(l−2)(n0−1).
When we evaluate this polynomial at the special point x, all the terms but the one with S=S1 vanish and we obtain
[TABLE]
Recall the leading term of Fτρ is xμ+λ with μ+λ=qs(m0+1,n0−1,m+1,l−1,n−1,τl).
Thus, there is a direct relationship between factorizations of Fτρ(x), which are highest weight Jack polynomials, and Jμ(x), which is a singular nonsymmetric Jack polynomial.
We finish this case with an example. Let N=8, κ=−2/3 and μ=(6,6,4,4,2,2,0,0). Then, τ=(24), λ=(3,3,2,2,1,1,0,0) and x=(z1,z2,y3,y3,y2,y2,y1,y1), for which
[TABLE]
Moreover, Fτρ is a specialization of the symmetric Jack polynomial Jλ+μ=J(9,9,6,6,3,3,0,0) to κ=−2/3.
6.2. The Hecke algebra case
There is a similarity with the symmetric group case. However, the difficulty arises because in this case there are factors of the form tuxi−xj, where the exponent u is related to a fixed tableau.
Our starting point is the same as for the symmetric group case: the polynomials {hS} of minimum degree for a given isotype τ that were defined in (3) and (4), together with the basis of isotype τ, {gS:S∈Tabτ} defined in Definition 2.2.
The property gSTi=−gS, for S such that colS[i]=colS[i+1] has an important consequence that we describe in the following result.
Proposition 6.5**.**
Let S∈Tabτ be such that colS[i]=colS[i+1]. Then, gS(x) is divisible by (txi−xi+1) and the quotient txi−xi+1gS(x) is symmetric in xi and xi+1.
Furthermore for i<j≤m such that colS[j]=colS[i], gS(x) is divisible by i≤j<k≤m∏(txj−xk).
Proof.
The equation gSTi=−gS is equivalent to txi−xi+1gS(x)=txi+1−xigS(xsi);, which is symmetric in xi and xi+1. Moreover, txi−xi+1gS(x)(txi+1−xi)=gS(xsi) is a polynomial, and thus by the unique factorization property gS(x) is divisible by txi−xi+1.
If gS(x) is divisible by (txj−xk) and k+1≤m, then the symmetry of txk−xk+1gS(x) under sk shows that gS(x) is divisible by (txj−xk+1). A similar argument applies to the situation in which i≤j<k−1≤m−1, and therefore, it shows that gS(x) is divisible by (txj+1−xk). Argue inductively first for k=i+1,…,m to demonstrate the factors (txi−xk) and then show that (txi+1−xk),(txi+2−xk),… are factors.
∎
Let us describe a modified version of the alternating polynomials defined for the symmetric group case.
Definition 6.6**.**
Let S∈Tabτ and (i,j) be a pair such that i<j and colS[i]=colS[j]. We denote by RS(i,j) the number of k>i such that rowS[k]=rowS[j] and colS[k]≥colS[i]. The modified alternating polynomials are defined as
[TABLE]
For example, consider S=\scalebox0.7\leavevmodeto71.53pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto42.67914pt0.0pt\pgfsys@moveto42.67914pt0.0pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto56.90552pt14.22638pt\pgfsys@lineto56.90552pt0.0pt\pgfsys@closepath\pgfsys@moveto56.90552pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto56.90552pt0.0pt\pgfsys@moveto56.90552pt0.0pt\pgfsys@lineto56.90552pt14.22638pt\pgfsys@lineto71.1319pt14.22638pt\pgfsys@lineto71.1319pt0.0pt\pgfsys@closepath\pgfsys@moveto71.1319pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke6\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.047.29233pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.061.5187pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture. Then, RS(3,6)=3 and then fS(x)=t3x3−x6. Observe that for S0 we have that RS0(i,j)=1, for all i<j such that colS0[i]=colS0[j], and then fS0(x)=hS0(x).
Note that fS(x)=aS(x) in the limiting symmetric group case t⟶1.
The following definition is very useful for our study.
Definition 6.7**.**
For S∈Tabτ, we say that (S,S(i)) is an adjacent pair if rowS[i]<rowS[i+1].
The following result shows how fS(i) can be computed by using fS for adjacent pairs since RS(i+1,u)=b and RS(i)(i,u)=b+1, with b=colS[i]−colS[i+1] and u=S(rowS[i],colS[i+1]).
Lemma 6.8**.**
Let S∈Tabτ be such that (S,S(i)) is an adjacent pair. Let b=colS[i]−colS[i+1] and u be entry in the position (rowS[i],colS[i+1]), that is u=S(rowS[i],colS[i+1]). Then,
[TABLE]
Now we are ready to introduce the specialization, which in this case depends on Y∈RSTτ.
Definition 6.9**.**
For the free variables y1, y2, …, yl−1, and z1, …, zτl, we define the special point at Y∈RSTτ by
[TABLE]
For example, consider Y=\scalebox0.7\leavevmodeto43.08pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture .Then, x(Y)=(z1,t−2y1,t−1y1,z2,y1).
We point out that the various polynomials fS, gS, and hS are indexed by S∈Tabτ, while the special points are labeled by Y∈RSTτ. Recall also that Tabτ⊂RSTτ. Moreover, if Y∈RSTτ and rowY[i]=rowY[i+1], then Y(i)∈RSTτ and x(Y(i))=x(Y)si.
Here is graphical representation of the idea. We built Tτ, for τ∈Par(N) by filling the boxes of τ in the following way: Tτ(i,j)=t1−jyi, for 1≤i<l and 1≤j≤τi, and Tτ(l,j)=zτl+1−j, for 1≤j≤τl. For instance, for τ=(4,2,2),
Then, x(Y)k=Tτ(rowS[k],colS[k]), for 1≤k≤N. Notice that Tτ is the same for all Y∈RSTτ and that the special point is a permutation of the list of entries of Tτ.
We demonstrate vanishing properties of type hS(x(Y))=0, for Y∈RSTτ, by using inductive techniques starting with hS0. The basic step relies on the transformation for the adjacent pair (S,S(i)):
[TABLE]
For the purpose of evaluating this formula at x(Y), note that x(Y)i=x(Y)i+1.
Proposition 6.10**.**
Consider S∈Tabτ such that (S,S(i)) is an adjacent pair, and Y∈RSTτ. If hS(x(Y))=0 and rowY[i]=rowY[i+1], then hS(i)(x(Y))=0.
Proof.
If rowY[i]<l, then x(Y)i+1=tx(Y)i by definition, and both terms in (14) vanish.
If rowY[i]=l, then hS(x(Y))=0 is an identity in the free
variables z1,z2,…,zτl. Moreover, this identity is thus invariant under any permutation of these variables, and so hS(x(Y)si)=0 in (14).
∎
For RSTτ, the induction is based on an inversion count. For Y∈RSTτ, define an inversion statistics as
inv0(Y):=#{(i,j)∣i<j and rowY[i]<rowY[j]}. For instance, take τ=(3,2). For Y=\scalebox0.7\leavevmodeto43.08pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture we have inv0(Y)=4 coming from the pairs (1,3), (1,4), (2,3), and (2,4). For Y1=\scalebox0.7\leavevmodeto43.08pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture we have inv0(Y1)=0.
Let us see some direct properties of inv0(Y).
Lemma 6.11**.**
Let Y∈RSTτ.
•
If rowY[i]=rowY[i+1], then inv0(Y(i))=inv0(Y)±1.
•
For Y=S1, inv0(Y)=0.
•
If (Y,Y(i)) is an adjacent pair, then inv0(Y(i))=inv0(Y)−1.
We prove the inductive step first, which is relatively easy.
Proposition 6.12**.**
Consider S∈Tabτ such that (S,S(i)) is an
adjacent pair. If hS(x(Y))=0 for each Y∈RSTτ such
that Y=S and inv0(Y)≤inv0(S), then hS(i)(x(Y′))=0 for each Y′∈RSTτ such that Y′=S(i) and inv0(Y′)≤inv0(S(i)).
Proof.
Consider Y′∈RSTτ satisfying the hypothesis of the statement. Then, Y′(i)=S (or else Y′=S(i)) and Y′=S because inv0(Y′)≤inv0(S)−1. Hence, hS(x(Y′))=0.
Now, if rowY′[i]=rowY′[i+1], then hS(i)(x(Y′))=0 by Proposition 6.10.
Otherwise Y′si∈RSTτ and inv0(Y′si)=inv0(Y′)±1≤inv0(S(i))+1=inv0(S). Thus, hS(x(Y′)si)=hS(x(Y′(i)))=0 and hS(i)(x(Y′))=0 by (14).
∎
It remains to show the basis of the induction, that is S0 satisfies the hypotheses of the Proposition 6.12.
Theorem 6.13**.**
Consider Y∈RSTτ such that Y=S0 and inv0(Y)≤inv0(S0). Then hS0(x(Y))=0.
Proof.
For 1≤n≤τl, let In denote the set of the entries in the nth column of S0. Then, we write Y as the disjoint union Y=j⋃(Y∩Ij), where each Y∩Ij is row-ordered skew shape. That is, for a,b∈Y∩Ij, a<b and rowY[a]=rowY[b] imply that colY[a]>colY[b]. Let C(i,j):={k∣k∈Y∩Ij and rowY[k]=i}. Since each entry in In is larger than each entry in Im, for n<m, and by the row-strictness of Y, each nonempty set C(i,j) consists of contiguous entries. Moreover, when we fix i, the sets C(i,1), C(i,2), … are not interlaced.
If for some i<l, there exists a j such that C(i,j) has at least two elements, then hS0(x(Y))=0. That is because there are two adjacent entries k1 and k2, with k1>k2, that are in the ith row of Y and in the jth column of S0. Therefore, hS0 has (txk2−xk1) as a factor and it vanishes at x(Y) because x(Y)k1=t⋅x(Y)k2.
Now suppose hS0(x(Y))=0. Then, by the previous argument, C(i,j) has at most one element for each i<l. Moreover, for 1≤k≤τl, Y∩Ik has size l since these are exactly the columns that extend to the bottom of the tableau. Then, C(l,k) cannot be empty (otherwise C(i,k) would have more than one element for some i<l), and C(l,k) cannot have more than one element (otherwise C(l,k′) would be empty for some other k′≤τl). Therefore, for 1≤i≤l, C(i,k) has exactly one element.
Our next step is to prove by induction on j that the jth column of Y is a permutation of Ij. Suppose that the first k−1 columns of Y are permutations of I1, …, Ik−1. Then, Y∩Ik⊂{[i,j]∣j≥k and 1≤i≤τj′≤τk′}. Since C(i,k) has at most one element and i=1∑τk′#C(i,k)=#Ik=τk′, it follows that C(i,k) has exactly one element, also for 1≤i≤τk′. In fact, C(i,k)=Y(i,k) or else C(i,k)=Y(i,m) for some m>k. Then, Y(i,k)<Y(i,m), violating the row-strictness of Y.
As a consequence S0 is obtained from Y by arranging each column in descending order, and thus inv0(Y)>inv0(S0), unless Y=S0.
Thus, hS0(x(Y))=0 implies Y=S0 or inv0(Y)>inv0(S0).
∎
For example, for τ=(3,3), S0=\scalebox0.7\leavevmodeto43.08pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt14.22638pt\pgfsys@moveto28.45276pt14.22638pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto42.67914pt28.45276pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@closepath\pgfsys@moveto42.67914pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke6\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture and inv0(S0)=3. Moreover, hS0(x)=(tx1−x2)(tx3−x4)(tx5−x6). Now, consider Y=\scalebox0.7\leavevmodeto43.08pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt14.22638pt\pgfsys@moveto28.45276pt14.22638pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto42.67914pt28.45276pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@closepath\pgfsys@moveto42.67914pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke6\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture, with inv0(Y)=5, and associated special point x(Y)=(y1t−2,z1,y1t−1,z2,z1,y1). When we evaluate hS0 at this special point, we get that hS0(x(Y))=0. However, if we consider Y′=\scalebox0.7\leavevmodeto43.08pt\vboxto28.85pt\pgfpicture\makeatletter\lower-0.2ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto0.0pt14.22638pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto14.22638pt0.0pt\pgfsys@closepath\pgfsys@moveto14.22638pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt0.0pt\pgfsys@moveto14.22638pt0.0pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto28.45276pt0.0pt\pgfsys@closepath\pgfsys@moveto28.45276pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt0.0pt\pgfsys@moveto28.45276pt0.0pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@lineto42.67914pt0.0pt\pgfsys@closepath\pgfsys@moveto42.67914pt14.22638pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto0.0pt14.22638pt\pgfsys@moveto0.0pt14.22638pt\pgfsys@lineto0.0pt28.45276pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto14.22638pt14.22638pt\pgfsys@closepath\pgfsys@moveto14.22638pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto14.22638pt14.22638pt\pgfsys@moveto14.22638pt14.22638pt\pgfsys@lineto14.22638pt28.45276pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto28.45276pt14.22638pt\pgfsys@closepath\pgfsys@moveto28.45276pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@moveto28.45276pt14.22638pt\pgfsys@moveto28.45276pt14.22638pt\pgfsys@lineto28.45276pt28.45276pt\pgfsys@lineto42.67914pt28.45276pt\pgfsys@lineto42.67914pt14.22638pt\pgfsys@closepath\pgfsys@moveto42.67914pt28.45276pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke6\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke4\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt3.89098pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke3\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.04.61319pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke5\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.018.83957pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.033.06595pt18.11736pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture, for which inv0(Y′)=2, we get that x(Y′)=(z1,z2,y1t−2,y1t−1,z3,y1) and that hS0(x(Y))=0.
Corollary 6.14**.**
Let S∈Tabτ and Y∈RSTτ be such that Y=S and inv0(Y)≤inv0(S). Then, hS(x(Y))=0.
Proof.
Since every S∈Tabτ can be reached from S0 by a sequence of adjacent pair constructions with a total number of inv0(S0)−inv0(S) steps, the conclusion follows from Theorem 6.13 and Proposition 6.12.
∎
Remark 6.15**.**
If we take the limit t⟶1 and set S=S1, we prove the corresponding result for the symmetric case, Theorem 6.3.
Since Tabτ⊂RSTτ, we can consider special points for S∈Tabτ, for which the specializations are also interesting.
Theorem 6.16**.**
For S∈Tabτ then hS(x(S))=fS(x(S)).
Proof.
We proceed by induction on the inversion number. The statement is trivially
true for S=S0. Since every S∈Tabτ can be reached from S0 by a sequence of adjacent pair constructions, assume that hS(x(S))=fS(x(S)) for some S such that (S,S(i)) is an adjacent pair.
Since inv0(S(i))=inv0(S)−1, by Theorem 6.14, hS(x(S(i)))=0. Now, we evaluate (14) at x(S(i))=x(S)si.
[TABLE]
Let i=S(r1,c1) and i+1=S(r2,c2) and u=S(r1,c2). By hypothesis r1<r2 and c1>c2, and x(S)i=t1−c1yr1 because r1<r2≤l.
By Lemma 6.8,
[TABLE]
Since x(S(i))u=t1−c2yr1 and x(S(i))i=x(S)i+1, we find that
[TABLE]
which completes the proof.
∎
Remark 6.17**.**
Again, taking the limit t⟶1, we prove the corresponding result for the symmetric case stated in Corollary 6.4.
We are ready to describe hS1(x(S1)), for which x(S1)i=zi, for 1≤i≤τl.
Proposition 6.18**.**
Set E(τ):=21i=1∑l(i−1)τi(τi−1). Then
[TABLE]
Proof.
The first equality comes from Theorem 6.16. For the second, we look at the terms in fS1(x(S1)) coming from the kth column, for 1≤k≤τl. For 1≤i<j<l, the entries from x(S1) corresponding to S1(i,k), S1(j,k) and S1(l,k) are t1−kyi, t1−kyj and zτl+1−k, respectively. Moreover, in S1 every entry in the ith row is larger than any entry in higher numbered rows, and therefore the exponent RS1(S1(i,k),S1(j,k))=τi−k+1. Putting this together, the corresponding factors in fS1(x(S1)) are
(tτi−k+1t1−kyj−t1−kyi), with 1≤k≤τj, and (tτi−k+1zτl+1−k−t1−kyi), with 1≤k≤τl. Thus
[TABLE]
Collecting the t1−k factors and using the sum k=1∑τj(1−k)=−21τj(τj−1) we obtain the term t−E(τ). Also, change the index k to τl+1−k in the first term.
∎
For example, take τ=(4,3,2,2) and
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[TABLE]
6.2.1. The general polynomials of isotype τ
In this section, the previous specialization and factorization results will be extended to arbitrary polynomials of isotype τ. Specifically, we consider polynomials with property SMP. The factorization and vanishing properties will be combined with the known factorizations of highest weight symmetric polynomials by means of
the projection map to obtain expressions consisting of purely linear factors.
We start by applying our previous results. Let {gS(x):S∈Tabτ} be a basis for a space of polynomials of isotype τ. By Proposition 6.5, gS0(x) is divisible by hS0(x) and {gS} satisfies the same transformation properties as {hS}, in particular the one described in (14). Finally, Proposition 6.12 and Theorem 6.13 also apply to {gS}, and so we have the following result.
Theorem 6.19**.**
Let S∈Tabτ and Y∈RSTτ such that Y=S and inv0(Y)≤inv0(S). Then gS(x(Y))=0.
Furthermore, if S=S1, then inv0(S)=0 and gS(x(S1))=0.
The analog of Theorem 6.16 follows using the same argument.
Theorem 6.20**.**
For S∈Tabτ,
[TABLE]
Note the specific formula for gS1(x(S1))=fS0(x(S0))gS0(x(S0))⋅fS1(x(S1)), which follows from Proposition 6.18. Thus, a factorization of fS0(x(S0))gS0(x(S0)) provides one for gS1(x(S1)), or conversely.
Here is an example in terms of some singular Macdonald polynomials. The polynomial M(2,2,2,0,0,0) is singular for qt2=−1 and of isotype (3,3). This polynomial corresponds to S1 while M(2,0,2,0,2,0), with qt2=−1, corresponds to S0, since α(S0)=(2,0,2,0,2,0) and α(S1)=(2,2,2,0,0,0). In this case, the special points are x(S0)=(z1,y1t−2,z2,y1t−1,z3,y1) and x(S1)=(z1,z2,z3,y1t−2,y1t−1,y1). Also,
[TABLE]
By computing Mα (using the Yang-Baxter graph, [14], for instance) and specializing it at q=−t−2, we get that
[TABLE]
Although, in general, there is no linear factorization of fS0(x(S0))gS0(x(S0)) necessarily.
6.2.2. Application to projections and factorizations
Let us consider the polynomials Mμ with
[TABLE]
with n≥2, 1≤νK≤n−1 and N=ℓ(μ)=dn−1+(n−1)(K−1)+νK. The associated isotype is τ=(dn−1,(n−1)K−1,νK). We also consider the specialization of the parameters q and t, ϖ=(q,t)=(ωu−n/g,um/g) where g=gcd(m,n) and ωm/g is a primitive gth root of unity.
In Section 5.4, we show that for quasistaircase partitions and their associated τ, there is a linear space of singular polynomials of isotype τ for HN(t), {Mα(S)∣S∈Tabτ}.
For S1, we use the following notation ν0=N and νi+1=N−(d−1)n−i(n−1), for 1≤i<K, so then α(S1)j=0 for ν1<j≤N, and α(S1)=(d+i−1)m, for νi+1<j≤νK.
Definition 6.21**.**
For free variables (z1,…,zνK,y1,…,yK), we define the special point associated to S1 by xj:=x(S1)j=tj−νi+1−1yK−i, for νi+1<j≤νi, and zj, otherwise.
For instance, take parameters N=11, d=2, n=3 and m=2. Then, μ=(8,8,6,6,4,4,05), τ=(5,2,2,2), [ν0,ν1,ν2,ν3]=[11,6,4,2], and x=(z1,z2,y1,ty1,y2,ty2,y3,ty3,t2y3,t3y3,t4y3).
For the projection application, let {hS:S∈Tabτ} denote the basis of minimal degree polynomials of isotype τ for HN(1/t). The minimal polynomials correspond to reverse lattice permutation of λ=((K−1)νK,(K−2)n−1,(K−3)n−1,…,1n−1,0dn−1).
We show that Fτρ=S∈Tabτ∑γ(S;t)1hSMα(S) is a highest weight symmetric Macdonald polynomial of index λ+μ, for parameters ϖ. Now, we evaluate Fτρ at the special point x. By Theorem 6.19, Mα(S)(x)=0 for S=S1, and therefore
[TABLE]
Notice that we do not claim that hS(x)=0 for S=S1. In fact, this does not hold because hS is associated with HN(1/t). However, since hS1 is symmetric for the variables in each row of S1, the evaluation formula applies with t replaced by 1/t in x because the substring [t−ky,t1−ky,…,y] is transformed to [tky,tk−1y,…,y] which is a permutation of u=[tky′,tk−1y′,…y′] where y′=tky, and u equals the appropriate substring of x with a changed variable. In fact, if u comes from the ith row of S1 and k=τi−1, then yi is replaced by yi′=tτi−1yi. This procedure leads to the desired evaluation of hS1(x). We also need to use the partial factorization of Mμ(x), proved in the previous section, since Mμ is of the form gS1 when (q,t)=ϖ.
In the paper [9, Theorem 7.3], we provide a linear factorization of Fτρ(x). To state the result introduce the product
[TABLE]
and for the parameters (m,n,K,νK;q,t), define
[TABLE]
The last product has the y-indices reversed from the original statement [9, Theorem 7.3]. That
formula is not symmetric in (y1,…,yK−2), for general (q,t), unless qmtn=1. Here the purpose is to demonstrate the role of hS1(x)hS1(x) in the factorization. Neither of hS1 and hS1 are symmetric in (y1,…,yK−1) but the product of them is. The result [9, Theorem 7.3] translates as Fτρ(x)=(∗)G(m,n,K,νK;q,t;z,y).
Our next goal is deriving formulas for hS1(x) and hS1(x).
Proposition 6.22**.**
With parameters (N,m,n,d) and (K,νK) and (q,t)=ϖ as above,
[TABLE]
Proof.
Since xj=x(S1)j, we apply the factorization in Proposition 6.18 to hS1(x). Moreover, rename the variables yi in that formula by yi′, so that we can do the change of coordinates yK=t2−dny1′ and yi=t2−nyK+1−i′, for 1≤i≤K−1. Furthermore, in this case, the parameters are l=K+1,τl=νK, τ1=dn−1 and τi=n−1, for 2≤i≤K. Therefore, the formula becomes
[TABLE]
Substituting qm=t−n in the claimed formula, this agrees up to a power of t with the latter formula.
∎
Proposition 6.23**.**
With parameters (N,m,n,d) and (K,νK) and (q,t)=ϖ as above,
[TABLE]
Proof.
First, we consider the changing the variables yi→t2−nyi, for 1≤i≤K−1, and yK→t2−dnyK. and so
[TABLE]
Now, we change t→t−1 to get
[TABLE]
This agrees up to a power of t with the stated formula, where the index s, with 0≤s≤n−2, is changed to k=s+1, so that 1≤k≤n−1.
∎
Recall that {Mα(S)∣S∈Tabτ} is a basis for isotype τ polynomials when (q,t)=ϖ. Then, as a consequence of the formula for Fτρ(x(S1)) from [9] and (15), we derive the following complete factorization for any S∈Tabτ.
Corollary 6.24**.**
[TABLE]
where we also have that
[TABLE]
7. Conclusion and perspective
By appealing to the theory of vector-valued Jack and Macdonald polynomials, we have demonstrated the close relationship between highest weight symmetric polynomials and nonsymmetric singular polynomials. The relationship depends entirely on certain representations of the symmetric groups or of the Hecke algebra. Picturesquely, these representations come from tableaux formed by stacking the steps of the quasistaircase on top of each other.
In previous work, the authors have studied clustering properties of symmetric Macdonald polynomials, that is, the factorization into linear factors of such a polynomial specialized to certain sets of points. These points typically have a number of free variables. The present work should be useful in finding and proving clustering properties of nonsymmetric Macdonald polynomials by
exploiting the projection relationship.
Clustering properties of certain (symmetric homogeneous) Jack polynomials are of great interest for physicist in particular in the Quantum Hall Effect theory [3, 4]. The tools developed in this paper as well as in the previous one [9] contribute to a better knowledge of the properties highlighted by Bernevig and Haldane. It is worth noting that these properties are illuminated when studying generalized versions of Jack’s polynomials: nonsymmetric, shifted, and q-deformed. In this paper, we have also used a more general variety of polynomials whose coefficients belong to irreducible representations of the Hecke algebra: the vector valued Macdonald polynomials. This shows that there is still a need to develop an arsenal of theoretical tools in order to fully understand the observations of physicists and in particular if we want to be able to properly prove the two remaining conjectures of Bernevig and Haldane. This is a broad research program that we will continue to explore in future works. It is very likely that the vector-valued Macdonald polynomials will play a central role.
One of our goals is to promote, among physicists, the use of q-deformations and vector-valued polynomials. If we could convince them that these generalizations allow us to understand the fine properties of the wave functions they investigate, then we would consider that we would have succeeded in our mission.
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The reference list from the paper itself. Each links out to its DOI / PubMed record.
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2[2] H. Belbachir, A. Boussicault, and J.-G. Luque. Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde. J. Algebra , 320(11):3911–3925, 2008.
3[3] B. A. Bernevig and F.D.M. Haldane. Model fractional quantum hall states and Jack polynomials. Phys. Rev. Lett. , 100:246802, 2008.
4[4] B. A. Bernevig and F.D.M. Haldane. Clustering properties and model wave functions for non-abelian fractional quantum hall quasielectrons. Phys. Rev. Lett. , 103, 2009.
5[5] A. Boussicault and J.-G. Luque. Staircase Macdonald polynomials and the q 𝑞 q -discriminant. In 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) , Discrete Math. Theor. Comput. Sci. Proc., AJ, pages 381–392. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008.
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