On some properties of symplectic Grothendieck polynomials
Eric Marberg
HKUST
[email protected]
The first author was supported by Hong Kong RGC Grant ECS 26305218.
Brendan Pawlowski
University of Southern California
[email protected]
Abstract
Grothendieck polynomials, introduced by Lascoux and Schützenberger,
are certain K-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the K-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the K-theoretic Schur P-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain “Grassmannian” orbit closures.
Contents
-
1 Introduction
-
2 Preliminaries
-
2.1 Permutations
-
2.2 Divided difference operators
-
2.3 Grothendieck polynomials
-
2.4 Symplectic Grothendieck polynomials
-
3 Transition equations
-
3.1 Lenart’s transition formula
-
3.2 Fixed-point-free Bruhat order
-
3.3 Symplectic transitions
-
4 Stable Grothendieck polynomials
-
4.1 K-theoretic Schur functions
-
4.2 Stabilization
-
4.3 K-theoretic Schur P-functions
-
4.4 Grassmannian formulas
1 Introduction
Let n be a positive integer.
The K-theory ring of the variety Fln of complete flags in Cn is isomorphic to a quotient of a polynomial ring [11, §2.3].
Under this correspondence, the Grothendieck polynomials Gw represent the classes of the structure sheaves of Schubert varieties. The results in this paper concern a family of symplectic Grothendieck polynomials GzSp which similarly represent the K-theory classes of the orbit closures of the complex symplectic group acting on Fln.
The Grothendieck polynomials Gw lie in Z[β][x1,x2,…], where β,x1,x2,… are commuting indeterminates,
and are indexed by elements w of the group S∞ of permutations of the positive integers P:={1,2,3,…} with finite support. Lascoux and Schützenberger
first defined these polynomials in a slightly different form in [13, 15]. Setting β=0
transforms Grothendieck polynomials to Schubert polynomials, which represent the Chow classes of Schubert varieties.
Lenart [17], extending work of Lascoux [14], proved a “transition formula” expressing any product xkGw as a finite linear combination of Grothendieck polynomials; the Bruhat order on S∞ controls which terms appear.
A nice corollary of Lenart’s result is that the set of Grothendieck polynomials form a Z[β]-basis
for the polynomial ring Z[β][x1,x2,…] (see Corollary 3.3).
For w∈S∞ and m≥0, let 1m×w∈S∞ denote the permutation sending i↦i for i≤m and m+i↦m+w(i) for i>m. The stable Grothendieck polynomial of w∈S∞ is then given by
[TABLE]
Results of Fomin and Kirillov [4] show that this limit converges in the sense of formal power series to a well-defined symmetric function.
Despite its name, Gw is a power series rather than a polynomial.
Of particular interest are the stable Grothendieck polynomials Gλ:=Gwλ where wλ is the Grassmannian permutation associated to an integer partition λ (see §4.1). The Gλ’s represent structure sheaves of Schubert varieties in a Grassmannian [2] and are natural “K-theoretic” generalizations of Schur functions. One can deduce from the transition formula for Gw that Gw is an N[β]-linear combination of Gλ’s, and the Hecke insertion algorithm of [3] leads to a combinatorial description of the coefficients in this expansion.
The symplectic Grothendieck polynomials GzSp are a second family of polynomials in Z[β][x1,x2,…], which now represent the K-theory classes of the orbit closures of the complex symplectic group acting on Fln for even n. They are indexed by elements z of the set I∞FPF of bijections z:P→P such that z=z−1, z(i)=i for all i∈P, and z(2i−1)=2i for all sufficiently large i.
(We think of I∞FPF as the set of fixed-point-free involutions of the positive integers with “finite support.”)
Wyser and Yong first considered these polynomials in [23], but their definition differs from ours by a minor change of variables. Setting β=0 gives the fixed-point-free involution Schubert polynomials studied in [6, 9, 23].
Our first main result, Theorem 3.8, is an analogue of Lenart’s transition formula for symplectic Grothendieck polynomials. This is somewhat more complicated than Lenart’s identity, involving multiplication of GzSp by two indeterminates xk and xz(k);
the corresponding proof is also more involved.
Nevertheless, there is a surprising formal similarity between the two transition equations.
A variant of Bruhat order again plays a key role.
This paper is a sequel to [20], where we showed that the
natural analogue of the stable limit (1.1) for symplectic Grothendieck polynomials
defines a symmetric formal power series GPzSp for each z∈I∞FPF.
Results of the first author [19] show that GPzSp is a finite N[β]-linear combination of
Ikeda and Naruse’s K-theoretic Schur P-functions GPλ [10].
Here we prove an important related fact: each GPλ occurs as GPzλSp where zλ∈I∞FPF is the FPF-Grassmannian involution corresponding to λ. See Theorem 4.17 for the precise statement.
Every symmetric power series in Z[β][[x1,x2,…]] can be written as a possibly infinite Z[β]-linear combination of stable Grothendieck polynomials.
One application of the preceding paragraph
is a proof that
each K-theoretic Schur P-function GPλ is a finite sum of Gμ’s with
coefficients in N[β].
It is also possible to deduce this fact from the results in [5, 22],
though the derivation is slightly roundabout;
see the remark after Corollary 4.18.
A brief outline of the rest of this article is as follows.
Section 2 covers some background material on permutations, divided difference operators,
and Grothendieck polynomials.
In Section 3 we review Lenart’s transition formula for Gw and then prove its symplectic analogue.
Section 4, finally, contains our results on symplectic stable Grothendieck polynomials.
2 Preliminaries
This section includes a few preliminaries and sets up most of our notation.
We write N={0,1,2,…} and P={1,2,3,…} for the sets of nonnegative and positive integers,
and define [n]:={1,2,…,n} for n∈N.
Throughout,
the symbols β, x1, x2, … denote commuting indeterminates.
2.1 Permutations
For i∈P, define si=(i,i+1) to be the permutation of P interchanging i and i+1.
These simple transpositions generate the infinite Coxeter group
S∞:=⟨si:i∈P⟩ of permutations of P with finite support,
as well as the finite subgroups Sn:=⟨s1,s2,…,sn−1⟩ for each n∈P.
The length of w∈S∞
is ℓ(w):=∣{(i,j)∈P×P:i<j and w(i)>w(j)}∣.
This finite quantity is also the minimum number of factors in any expression for w as a product of simple transpositions.
We represent elements of S∞ in one-line notation
by identifying a word w1w2⋯wn that has {w1,w2,…,wn}=[n]
with the permutation w∈S∞ that has w(i)=wi for i∈[n] and w(i)=i for all integers i>n.
2.2 Divided difference operators
Let L=Z[β][x1±1,x2±1,…] denote the ring of Laurent polynomials
in the variables x1, x2, … with coefficients in Z[β].
Given i∈P and f∈L,
write sif for the Laurent polynomial formed from f by interchanging the variables xi and xi+1.
This operation extends to a group action of S∞ on L.
For i∈P,
the divided difference operators ∂i and ∂i(β) are the maps L→L given by
[TABLE]
Both operators preserve the subring of polynomials Z[β][x1,x2,…]⊂L.
Some identities are useful for working with these maps.
All formulas involving ∂i(β)
reduce to formulas involving ∂i on setting β=0.
Fix i∈P
and f,g∈L.
Then
[TABLE]
and we have ∂if=0 and ∂i(β)f=−βf if and only if sif=f,
in which case
[TABLE]
Moreover, one has ∂i∂i=0 and ∂i(β)∂i(β)=−β∂i(β).
Both families of operators
satisfy the usual braid relations for S∞, meaning that we have
[TABLE]
for all i,j∈P with ∣i−j∣>1.
If w∈S∞ then we can therefore define
[TABLE]
where w=si1si2⋯sil is any reduced expression,
i.e., a minimal length factorization of w as a product of simple transpositions.
2.3 Grothendieck polynomials
The following definition of Grothendieck polynomials originates in [4].
Theorem-Definition 2.1** (Fomin and Kirillov [4]).**
There exists a unique family {Gw}w∈S∞⊂Z[β][x1,x2,…] with
Gn⋯321=x1n−1x2n−2⋯xn−11
for all n∈P
and such that
∂i(β)Gw=Gwsi for i∈P with w(i)>w(i+1).
Note that it follows that
∂i(β)Gw=−βGw if w(i)<w(i+1).
Example 2.2**.**
The Grothendieck polynomials for w∈S3 are
[TABLE]
We typically suppress the parameter β in our notation,
but for the moment write Gw(β)=Gw for w∈S∞.
The Schubert polynomial Sw of a permutation w∈S∞ (see [18, Chapter 2])
is then Gw(0). The polynomials {Sw}w∈S∞ are a Z[β]-basis for
Z[β][x1,x2,…] [18, Proposition 2.5.4] so the Grothendieck polynomials are linearly independent.
Some references use the term “Grothendieck polynomial” to refer to the polynomials Gw(−1).
One loses no generality in setting β=−1 since one can show by downward induction on permutation length that
[TABLE]
Thus, it is straightforward to translate formulas in Gw(−1) to formulas
in Gw(β).
2.4 Symplectic Grothendieck polynomials
Let Θ:P→P be the map
sending i↦i−(−1)i,
so that Θ=(1,2)(3,4)(5,6)⋯.
Define
I∞FPF:={w−1Θw:w∈S∞}.
The elements of I∞FPF are the involutions of the positive integers
that have no fixed points and that agree with Θ at all sufficiently large values of i.
We represent elements of I∞FPF in one-line notation by identifying
a word z1z2⋯zn, satisfying {z1,z2,…,zn}=[n] and
zi=j if and only if zj=i=j,
with the involution z∈I∞FPF that has z(i)=zi for i∈[n] and z(i)=Θ(i) for i>n.
The symplectic analogues of Gw introduced below
were first studied by Wyser and Yong in a slightly different form; see [23, Theorems 3 and 4].
The characterization given here combines [20, Theorem 3.10 and Proposition 3.11].
Theorem-Definition 2.3** ([20, 23]).**
There exists a unique family {GzSp}z∈I∞FPF⊂Z[β][x1,x2,…]
with
Gn⋯321Sp=∏1≤i<j≤n−i(xi+xj+βxixj)
for all n∈2P
and such that
∂i(β)GzSp=GsizsiSp for i∈P with
i+1=z(i)>z(i+1)=i.
The elements of this family are the symplectic Grothendieck polynomials
described in the introduction.
If i∈P is such that z(i)<z(i+1) or i+1=z(i)>z(i+1)=i
then ∂i(β)GzSp=−βGzSp [20, Proposition 3.11].
Example 2.4**.**
The polynomials for z∈I4FPF:={wΘw−1:w∈S4} are
[TABLE]
The smallest example of GzSp where z is not Sp-dominant
(see Theorem 2.5)
is
[TABLE]
Setting β=0 transforms GzSp to the fixed-point-free involution Schubert polynomials SzSp studied in [6, 7, 9, 23].
Since the family {SzSp}z∈I∞FPF is linearly independent,
{GzSp}z∈I∞FPF is also linearly independent.
We need one other preliminary result concerning the polynomials GzSp.
The symplectic Rothe diagram of an involution z∈I∞FPF is the set of pairs
[TABLE]
An element z∈I∞FPF is Sp-dominant if
DSp(z)={(i+j,j)∈P×[k]:1≤i≤μj}
for a strict partition μ=(μ1>μ2>⋯>μk>0).
This condition holds, for example, when z=n⋯321 for any n∈2P.
Theorem 2.5** ([20, Theorem 3.8]).**
If z∈I∞FPF is Sp-dominant then
[TABLE]
3 Transition equations
Lenart [17] derives a formula expanding the product xkGv for k∈P and v∈S∞ in terms of other Grothendieck polynomials.
In this section, we prove a similar identity for symplectic Grothendieck polynomials.
3.1 Lenart’s transition formula
We recall Lenart’s formula to motivate our new results.
Given v∈S∞ and k∈P, define Pk(v) to be the set of
all permutations in S∞ of the form
[TABLE]
where p,q∈N and
ap<⋯<a2<a1<k<bq<⋯<b2<b1,
and the length increases by exactly one
upon multiplication by each transposition.
Differing slightly from the convention in [17], we allow the case p=q=0 so w∈Pk(v).
Given w∈Pk(v) define ϵk(w,v)=(−1)p.
This notation is well-defined since
p can be recovered from w∈Pk(v) as the number of indices i<k with v(i)=w(i).
Theorem 3.1** ([17, Theorem 3.1]).**
If v∈S∞ and k∈P then
[TABLE]
The cited theorem of Lenart applies to the case when β=−1,
but this is equivalent to the given identity for generic β by (2.5).
Example 3.2**.**
Taking v=13452∈S∞ and k=3 in Theorem 3.1 gives
[TABLE]
This reduces to [17, Example 3.9] on setting β=−1.
Lenart’s formula implies that xkGv is a finite Z[β]-linear combination of Gw’s.
By starting with v=1 so that Gv=1, we deduce that any monomial
in Z[β][x1,x2,…] is a finite linear combination of Grothendieck polynomials.
Since these functions are also linearly independent, the following holds:
Corollary 3.3**.**
The set {Gw}w∈S∞ is a Z[β]-basis for Z[β][x1,x2,…].
Remark 3.4**.**
This corollary is nontrivial since Gw is an inhomogeneous polynomial of the form
\mathfrak{S}_{w}+(\text{ terms of degree greater than \ell(w) }).
Since {Sw}w∈S∞ is a Z-basis for Z[x1,x2,…],
it follows that any polynomial in Z[β][x1,x2,…] can be inductively expanded in terms of Grothendieck polynomials.
However, it is not clear a priori that such an expansion will terminate in a finite sum.
For v,w∈S∞, write v⋖w if ℓ(w)=ℓ(v)+1 and v−1w=(i,j) is a transposition for some positive integers i<j.
It is well-known that if w∈S∞ and i,j∈P are such that i<j,
then w⋖w(i,j) if and only if w(i)<w(j) and no integer e has i<e<j and w(i)<w(e)<w(j).
For distinct integers i,j∈P, let tij be the linear operator, acting on the right,
with Gwtij=Gw(i,j) for w∈S∞.
We can restate Theorem 3.1 as the following identity:
Theorem 3.5**.**
Fix v∈S∞ and k∈P.
Suppose
[TABLE]
are the integers such that v⋖v(j,k) and v⋖v(k,l).
Then
[TABLE]
Proof.
After setting β=−1,
this is a slight generalization of [17, Corollary 3.10] (which is the main result of [14]),
and has nearly the same proof.
Let J={j1,j2,…,jp} and L={l1,l2,…,lq}.
For subsets E={e1<e2<⋯<em}⊂J
and F={fn<⋯<f2<f1}⊂L
define tE,k,tk,F∈S∞ by
[TABLE]
One has ℓ(vtE,k)=ℓ(v)+∣E∣ and ℓ(vtk,F)=ℓ(v)+∣F∣
for all choices of E⊂J and F⊂L.
Hence, by Theorem 3.1, we must show that
[TABLE]
Each permutation w indexing the sum on the left can be written as
[TABLE]
for some indices with i1<i2<⋯<im>im+1>⋯>in and
in+1>⋯>ir>k
and {i1,i2,…,im}⊂J.
Here, the set indexing the outer sum
on the left side of (3.1) is E={i1,i2,…,im}.
If n>0 then each such w appears twice
with opposite associated signs ϵk(w,vtE,k); the two appearances correspond to
E={i1,…,im} and E={i1,…,im−1}.
The permutations w that arise with n=0, alternatively, are exactly
the elements vtk,F for F⊂L, so (3.1) holds.
∎
3.2 Fixed-point-free Bruhat order
For each involution z∈I∞FPF, let
[TABLE]
One can check that if z∈I∞FPF and i∈P then
[TABLE]
It follows by induction that
ℓFPF(z)=min{ℓ(w):w∈S∞ and w−1Θw=z}.
For y,z∈I∞FPF, we write
y⋖Fz
if ℓFPF(z)=ℓFPF(y)+1 and z=tyt for a transposition t∈S∞.
The transitive closure of this relation is the Bruhat order on I∞FPF from [9, §4.1].
One can give
a more explicit characterization of ⋖F:
Proposition 3.6** ([7, Proposition 4.9]).**
Suppose y∈I∞FPF, i,j∈P, and i<j.
- (a)
If y(i)<i then y⋖F(i,j)y(i,j) if and only if these properties hold:
Either y(i)<i<j<y(j) or y(i)<y(j)<i<j.
No integer e has i<e<j and y(i)<y(e)<y(j).
2. (b)
If j<y(j) then y⋖F(i,j)y(i,j) if and only if these properties hold:
Either y(i)<i<j<y(j) or i<j<y(i)<y(j).
No integer e has i<e<j and y(i)<y(e)<y(j).
Remark 3.7**.**
Let y∈I∞FPF and i<j and t=(i,j)∈S∞.
The cases when y⋖Ftyt correspond to the following pictures,
in which the edges indicate the cycle structure of the relevant involutions
restricted to {i,j,y(i),y(j)}:
[TABLE]
[TABLE]
[TABLE]
3.3 Symplectic transitions
For distinct i,j∈P,
define uij to be the linear operator
with GzSpuij=G(i,j)z(i,j)Sp for z∈I∞FPF.
One cannot hope for a symplectic version of Theorem 3.1
since products of the form (1+βxk)GzSp may fail to be linear combinations
of symplectic Grothendieck polynomials.
There is an analogue of Theorem 3.5, however:
Theorem 3.8**.**
Fix v∈I∞FPF and j,k∈P with v(k)=j<k=v(j).
Suppose
[TABLE]
are the integers such that v⋖F(i,j)v(i,j) and v⋖F(k,l)v(k,l).
Then
[TABLE]
is equal to
[TABLE]
This is a generalization of [7, Theorem 4.17],
which one recovers by subtracting GvSp from (3.5) and (3.6), dividing
by β,
and then setting β=0.
These results belong to a larger family of similar formulas
related to Schubert calculus; see also [1, 12, 21].
Before giving the proof, we present one example.
Example 3.9**.**
If v=(1,2)(3,5)(4,8)(6,7)∈I∞FPF and (j,k)=(3,5), then
we have {i1<i2<⋯<ip}={2} and {l1>l2>⋯>lq}={6,8}
and Theorem 3.8 is equivalent, after a few manipulations, to the claim that
(x3+x5+βx3x5)G(1,2)(3,5)(4,8)(6,7)Sp+(1+βx3)(1+βx5)G(1,3)(2,5)(4,8)(6,7)Sp
is equal to
G(1,2)(3,8)(4,5)(6,7)Sp+G(1,2)(3,6)(4,8)(5,7)Sp+βG(1,2)(3,8)(4,6)(5,7)Sp.
Proof of Theorem 3.8.
The proof is by downward induction on ℓFPF(v).
As a base case, suppose v=n⋯321∈I∞FPF where n∈2P, so that j=n+1−k.
Then p=0, q=1, l1=n+1, and the theorem reduces to the claim that
[TABLE]
for w:=(k,n+1)v(k,n+1).
This follows from
Theorem 2.5
since w is Sp-dominant with DSp(w)=DSp(v)⊔{(n+1−j,j)}.
Now let v∈I∞FPF∖{Θ} and j,k∈P be arbitrary with v(k)=j<k=v(j).
It is helpful to introduce some relevant notation. Define
[TABLE]
and let Asc−(v,j,k)={i1,i2,…,ip} and Asc+(v,j,k)={l1,l2,…,lq}
where the indices i1,i2,…,ip and l1,l2,…,lq are as in (3.4).
For each nonempty subset A={a1<a2<⋯<am}⊂Asc−(v,j,k), define
[TABLE]
For each nonempty subset B={bm<⋯<b2<b1}⊂Asc+(v,j,k), define
[TABLE]
For empty sets, we define τ∅±(v,j,k)=v.
It then follows from Proposition 3.6
that ℓFPF(τS±(v,j,k))=ℓFPF(v)+∣S∣ for all choices of S, and we have
[TABLE]
If we represent elements of I∞FPF as arc diagrams, i.e., as perfect matchings on the positive integers
with an edge for each 2-cycle,
then the elements τS±(v,j,k) can be understood as follows.
The arc diagram of τS−(v,j,k) is formed from v by cyclically shifting up the endpoints S⊔{j}. For example, if the relevant part of the arc diagram of v appears as
[TABLE]
where the elements of S⊔{j} are labeled by ∘ while the elements of v(S)⊔{k} are labeled by ∙, then the arc diagram of τS−(v,j,k) is
[TABLE]
Similarly,
the arc diagram of τS+(v,j,k) is formed from v by cyclically shifting down the endpoints {k}⊔S. For example, if the arc diagram of v is
[TABLE]
where the elements of S⊔{k} are labeled by ∘ while the elements of v(S)⊔{j} are labeled by ∙, then the arc diagram of τS−(v,j,k) is
[TABLE]
Suppose the theorem holds for a given v∈I∞FPF∖{Θ}
in the sense that
(1+βxj)(1+βxk)Π−(v,j,k)=Π+(v,j,k)
for all choices of v(k)=j<k=v(j).
Let d∈P be
any positive integer
with d+1=v(d)>v(d+1)=d
and set
[TABLE]
Choose integers j,k∈P with v(k)=j<k=v(j);
note that we cannot have j=d<d+1=k.
In view of the first paragraph, it is enough to show that
[TABLE]
where j′=sd(j) and k′=sd(k).
There are seven cases to examine:
Case 1: Assume that d+1<j.
We must show that
[TABLE]
It suffices by (2.3) to prove that
∂d(β)Π±(v,j,k)=Π±(w,j,k).
The + form of this claim is straightforward from Proposition 3.6 and (3.7);
in particular, it holds that Asc+(w,j,k)=Asc+(v,j,k).
For the other form, there are four subcases to consider:
Assume that d,d+1∈/Asc−(v,j,k).
Since v(d)>v(d+1), it is again straightforward from Proposition 3.6 and (3.7)
to show that Asc−(w,j,k)=Asc−(v,j,k) and ∂d(β)Π−(v,j,k)=Π−(w,j,k).
Assume that d∈Asc−(v,j,k) and d+1∈/Asc−(v,j,k).
Then
[TABLE]
and d+1=τS−(v,j,k)(d)>τS−(v,j,k)(d+1)=d for all subsets S⊂Asc−(v,j,k),
so we again have ∂d(β)Π−(v,j,k)=Π−(w,j,k).
Assume that d∈/Asc−(v,j,k) and d+1∈Asc−(v,j,k).
This can only occur if d<k<v(d), so we have
[TABLE]
From here, we deduce that ∂d(β)Π−(v,j,k)=Π−(w,j,k) by
an argument similar to the one in case (1b).
Assume that d,d+1∈Asc−(v,j,k).
Three situations are possible for the relative order of d, d+1, v(d), and v(d+1).
First suppose v(d+1)<d<d+1<v(d).
Then
[TABLE]
and every i∈Asc−(v,j,k) with i<d must have v(d)<v(i)<k.
It follows that if S⊂Asc−(v,j,k) then
[TABLE]
If we have v(d+1)<v(d)<d or j<v(d+1)<v(d)<k
then (3.8) and (3.9) both still hold and follow by similar reasoning.
Combining these identities with (3.7) gives ∂d(β)Π−(v,j,k)=Π−(w,j,k).
We conclude from this analysis that ∂d(β)Π±(v,j,k)=Π±(w,j,k).
Case 2: Assume that d+1=j, so that k<v(j−1).
We must show that
[TABLE]
It follows from (2.2) that
∂j−1(β)((1+βxj)(1+βxk)Π−(v,j,k))
is equal to
[TABLE]
and it is easy to see that
∂j−1(β)Π−(v,j,k)=Π−(w,j−1,k).
Thus, it suffices to show that
[TABLE]
It follows from Proposition 3.6 that
Asc+(w,j−1,k)={l∈Asc+(v,j,k):l<v(j−1)}⊔{v(j)}.
We deduce that if S⊂Asc+(v,j,k) then
[TABLE]
If v(j)∈S⊂Asc+(w,j−1,k) then
τS+(w,j−1,k)=τS∖{v(j)}+(v,j,k).
Combining these identities with (3.7) shows the needed claim (3.10).
Case 3: Assume that d=j,
so that either v(j+1)<j<j+1<k or j<j+1<v(j+1)<k.
We must show that
[TABLE]
It follows from (2.2) that
∂j(β)((1+βxj)(1+βxk)Π−(v,j,k))
is equal to
[TABLE]
It is easy to deduce that
∂j(β)Π+(v,j,k)=Π+(w,j+1,k) from Proposition 3.6,
so it suffices to show that
[TABLE]
First assume v(j+1)<j<j+1<k.
Then every i∈Asc−(v,j,k) with i<v(j+1)
must have j+1<v(i)<k, and Asc−(w,j+1,k) is equal to
[TABLE]
As in Case 2, we deduce that if S⊂Asc−(v,j,k) then
[TABLE]
On the other hand, if j∈S⊂Asc−(w,j+1,k) then
[TABLE]
Combining these identities with (3.7) gives (3.11) as desired.
Alternatively, if we have j<j+1<v(j+1)<k,
then
[TABLE]
and we deduce by similar reasoning that the identities (3.12) and (3.13) both still hold,
so (3.11) again follows.
Case 4: Assume that j<d and d+1<k.
We must show that
[TABLE]
It suffices by (2.3) to prove that
∂d(β)Π±(v,j,k)=Π±(w,j,k).
There are three subcases to consider:
- (4a)
If j<v(d)<k or j<v(d+1)<k or v(d+1)<j<k<v(d) then
the desired identities are straightforward from Proposition 3.6.
2. (4b)
Assume that v(d+1)<v(d)<j.
In this case it is easy to see that ∂d(β)Π+(v,j,k)=Π+(w,j,k)
and if v(d+1)∈/Asc−(v,j,k),
then
we likewise deduce that ∂d(β)Π−(v,j,k)=Π−(w,j,k).
Suppose instead that v(d+1)∈Asc−(v,j,k).
We then also have v(d)∈Asc−(v,j,k),
but no i∈Asc−(v,j,k) is such that v(d+1)<i<v(d),
and
[TABLE]
It follows that if S⊂Asc−(v,j,k) then
[TABLE]
Combining this with (3.7) gives ∂d(β)Π−(v,j,k)=Π−(w,j,k).
3. (4c)
Assume that k<v(d+1)<v(d).
This is the mirror image of (4b) and we get
∂d(β)Π±(v,j,k)=Π±(w,j,k) by symmetric arguments.
Case 5: Assume that d+1=k,
so that either j<k−1<k<v(k−1) or j<v(k−1)<k−1<k.
We must show that
[TABLE]
It follows from (2.2) that
∂k−1(β)((1+βxj)(1+βxk)Π−(v,j,k))
is equal to
[TABLE]
and it is easy to deduce that
∂k−1(β)Π−(v,j,k)=Π−(w,j,k−1).
It therefore suffices to show that
(∂k−1(β)+β)Π+(v,j,k)=Π+(w,j,k−1).
The required argument is the mirror image of Case 3; we omit the details.
Case 6: Assume that d=k.
We must show that
[TABLE]
It follows from (2.2) that
∂k(β)((1+βxj)(1+βxk)Π−(v,j,k)) is equal to
[TABLE]
and it is easy to see that
∂k(β)Π+(v,j,k)=Π+(w,j,k+1).
Thus, it suffices to show that
(∂k(β)+β)Π−(v,j,k)=Π−(w,j,k+1).
The required argument is the mirror image of Case 2; we omit the details.
Case 7: Finally, assume that k<d.
We must show that
[TABLE]
It suffices by (2.3) to prove that
∂d(β)Π±(v,j,k)=Π±(w,j,k).
The required argument is the mirror image of Case 1; we omit the details.
This case analysis completes our inductive proof.
∎
Corollary 3.10**.**
Suppose v∈I∞FPF and j,k∈P have j<k=v(j). Then
[TABLE]
Proof.
It follows by induction from Theorem 3.8
that (1+βxj)(1+βxk)GvSp is a possibly infinite Z[β]-linear combination of GzSp’s.
This combination must be finite by Corollary 3.3,
since no Grothendieck polynomial Gw appears in the expansion of GySp and GzSp for distinct y,z∈I∞FPF
by [20, Theorem 3.12].
∎
A visible descent of z∈I∞FPF is an integer i such that z(i+1)<min{i,z(i)}.
The following corollary is a symplectic analogue of Lascoux’s transition equations for Grothendieck polynomials in [14].
Corollary 3.11**.**
Let k∈P be the last visible descent of z∈I∞FPF.
Define l to be the largest integer with k<l and z(l)<min{k,z(k)},
and set
[TABLE]
Let 1≤i1<i2<⋯<ip<j be the integers with v⋖F(i,j)v(i,j). Then
[TABLE]
Note that one could rewrite the right side
without using any minus signs.
Proof.
It suffices by Theorem 3.8
to show that Asc+(v,j,k)={l}.
This is precisely [9, Lemma 5.2], but also follows as a self-contained exercise.
∎
4 Stable Grothendieck polynomials
The limit of a sequence of polynomials or formal power series is defined to converge if the coefficient sequence for any fixed monomial is eventually constant.
Given n∈N and w∈S∞,
write 1n×w∈S∞ for the permutation
that maps i↦i for i≤n and i+n↦w(i)+n for i∈P.
The stable Grothendieck polynomial of w∈S∞ is defined as the limit
[TABLE]
Remarkably, this always converges to a well-defined symmetric function [2, §2].
Given n∈N and z∈I∞FPF,
we similarly write (21)n×z∈I∞FPF for the involution
mapping i↦i−(−1)i for i≤2n and i+2n↦z(i)+2n for i∈P.
Following [20], the symplectic stable Grothendieck polynomial of z∈I∞FPF is defined as
[TABLE]
The next lemma is a consequence of [20, Theorem 3.12 and Corollary 4.7]:
Lemma 4.1** ([20]).**
The limit (4.2) converges for all z∈I∞FPF.
Moreover, the resulting power series GPzSp is
the image of GzSp under the linear map
Z[β][x1,x2,…]→Z[β][[x1,x2,…]] with Gw↦Gw
for w∈S∞.
It follows that
GPzSp is also a symmetric function.
These power series have some stronger symmetry properties, which we explore in this section.
4.1 K-theoretic Schur functions
Besides permutations and involutions, there is also a notion of stable Grothendieck polynomials
for partitions, though these would more naturally be called K-theoretic Schur functions.
The precise definition is as follows.
If λ=(λ1≥λ2≥⋯≥λk>0) is an integer partition, then a set-valued tableau of shape λ
is a map T:(i,j)↦Tij from the Young diagram
[TABLE]
to the set of finite, nonempty subsets of P.
For such a map T, define
[TABLE]
A set-valued tableau T is semistandard if one has max(Tij)≤min(Ti,j+1)
and max(Tij)<min(Ti+1,j) for all relevant (i,j)∈Dλ.
Let SetSSYT(λ) denote the set of semistandard set-valued tableaux of shape λ.
Definition 4.2** ([2]).**
The stable Grothendieck polynomial of a partition λ is
[TABLE]
This definition sometimes appears in the literature with the parameter β
set to ±1,
but if we write Gλ(β)=Gλ then
(−β)∣λ∣Gλ(β)=Gλ(−1)(−βx1,−βx2,…).
Setting β=0 transforms Gλ to the usual Schur function sλ.
For example, if λ=(1) then
G(1)=s(1)+βs(1,1)+β2s(1,1,1)+….
The functions Gλ are related to Gw for w∈S∞ by the following result of Buch [2].
For a partition λ with k parts,
define wλ∈S∞ to be the permutation
with wλ(i)=i+λk+1−i for i∈[k] and wλ(i)<wλ(i+1)
for all i>k.
Theorem 4.3** ([2, Theorem 3.1]).**
If λ is any partition then Gwλ=Gλ.
Write P for the set of all partitions.
Theorem 4.4** ([3, Theorem 1]).**
If w∈S∞ then
Gw∈N[β]-span{Gλ:λ∈P}.
A symplectic analogue of Theorem 4.4 is known [19, Theorem 1.9].
Our goal in the rest of this section is to prove a symplectic analogue of Theorem 4.3.
Remark**.**
Theorem 4.4 is a corollary of a stronger result [3, Theorem 1], which
gives a formula for the expansion of Gw into Gλ’s in terms of increasing tableaux.
Knowing this formula, one can recover
Theorem 4.3 by checking that there is a
unique increasing tableau whose reading word is a Hecke word for a Grassmannian permutation.
It may be possible to use a similar strategy to prove our symplectic analogue of Theorem 4.3
(given as Theorem 4.17) from [19, Theorem 1.9]. We present a different algebraic proof here, which is independent of [19].
4.2 Stabilization
We refer to the linear map Z[β][x1,x2,…]→Z[β][[x1,x2,…]] with Gw↦Gw
as stabilization.
It will be useful in the next two sections to have a description of this operation
in terms of divided differences.
As in Section 2.2, let L=Z[β][x1±1,x2±1,…].
For i∈P,
write πi(β) for the isobaric divided difference operator defined by the formula
[TABLE]
We have πi(β)f=f if and only if sif=f,
in which case πi(β)(fg)=f⋅πi(β)g.
These operators are idempotent with
πi(β)πi(β)=πi(β)
for all i∈P, and we have
[TABLE]
for all i,j∈P with ∣i−j∣>1.
For w∈S∞ we can therefore define
[TABLE]
where w=si1si2⋯sil is any reduced expression.
Given f∈Z[β][[x1,x2,…]] and n∈N,
write f(x1,…,xn) for the polynomial obtained by setting
xn+1=xn+2=⋯=0
and let wn=n⋯321∈Sn.
Proposition 4.5**.**
If v∈Sn then
G1N×v(x1,…,xn)=πwn(β)Gv for all N≥n.
Proof.
Fix v∈Sn and define τN:=π1(β)π2(β)⋯πN−1(β).
We then have
[TABLE]
Let u=(v1+1)(v2+2)⋯(vn+1)1∈Sn+1. Since
[TABLE]
it follows that τn+1Gv=G11×v and
G1N×v=τn+N⋯τn+2τn+1Gv for all N∈P.
Define rn(f):=f(x1,x2,…,xn).
Then (4.3) implies that
[TABLE]
so rn(τNf)=τnrN(f) for n≤N
and rn(τNf)=τNrn(f) for N<n.
Since rn(Gv)=Gv and
(τn)n=πwn(β),
we have rn(G1N×v)=πwn(β)Gv
for N≥n.
∎
Corollary 4.6**.**
If v∈S∞ and z∈I∞FPF then
[TABLE]
Proof.
These identities are clear from Lemma 4.1 and Proposition 4.5.
∎
For any polynomials x and y, let
[TABLE]
For integers 0<a≤b, define
[TABLE]
so that ∂a↘a=∂a↘a(β)=1.
Finally, let Δm,n(β)(x):=∏j=2n(1+βxm+j)j−1.
Lemma 4.7**.**
If m∈N and n∈P then
[TABLE]
where in the last sum Sn acts by permuting the variables xm+1,xm+2,…,xm+n.
Proof.
The second equality is [18, Proposition 2.3.2].
The first equality follows by induction: the base case when n=1 holds by definition, and if n>1 then
∂1m×wn(β)=∂(m+n)↘(m+1)(β)∂1m+1×wn−1(β)
and the desired identity is easy to deduce using the fact that
∂b↘a(β)f=∂b↘a((1+βxb)⋯(1+βxa+2)(1+βxa+1)f).
∎
For any integer sequence λ=(λ1,λ2,…) with finitely many
nonzero terms, define xλ:=x1λ1x2λ2⋯.
Let δn:=(n−1,n−2,…,2,1,0)
for n∈P.
Lemma 4.8**.**
If n∈P then πwn(β)f=∂wn(β)(xδnf) for all f∈L.
Proof.
The expression wn=(s1)(s2s1)(s3s2s1)⋯(sn−1⋯s3s2s1)
is reduced
and one can check, noting that ∂i(β)(x1x2⋯xnf)=x1x2⋯xn⋅∂i(β)f for i<n, that
∂n−1(β)⋯∂2(β)∂1(β)(xδnf)=xδn−1πn−1(β)⋯π2(β)π1(β)f. The lemma follows by induction from these identities.
∎
Corollary 4.9**.**
If λ is a partition then
Gλ=limn→∞πwn(β)(xλ).
Proof.
Apply Lemmas 4.7 and 4.8 to [10, Eq. (2.14)], for example.
∎
4.3 K-theoretic Schur P-functions
The natural symplectic analogues of Theorems 4.3
and 4.4 involve shifted versions of the symmetric functions Gλ,
which we review here.
Define the marked alphabet to be the totally ordered set
of primed and unprimed integers M:={1′<1<2′<2<…},
and write
∣i′∣:=∣i∣=i for i∈P.
If λ=(λ1>λ2>⋯>λk>0) is a strict partition, then a shifted set-valued tableau of shape λ
is a map T:(i,j)↦Tij from the shifted diagram
[TABLE]
to the set of finite, nonempty subsets of M.
Given such a map T, define
[TABLE]
A shifted set-valued tableau T is semistandard if
for all relevant (i,j)∈\SSλ:
max(Tij)≤min(Ti,j+1) and Tij∩Ti,j+1⊂{1,2,3,…}.
max(Tij)≤min(Ti+1,j) and Tij∩Ti+1,j⊂{1′,2′,3′,…}.
In such tableaux, an unprimed number can appear at most once in a column, while a primed number
can appear at most one in a row.
Let SetSSMT(λ) denote the set of semistandard shifted set-valued tableaux of shape λ.
Definition 4.10** ([10]).**
The K-theoretic Schur P-function
of a strict partition λ is the power series
GPλ:=∑Tβ∣T∣−∣λ∣xT
where the summation is over tableaux T∈SetSSMT(λ) with no primed numbers in any position on the main diagonal.
This definition is due to Ikeda and Naruse [10], who
also show that each
GPλ is symmetric in the xi variables
[10, Theorem 9.1].
Setting β=0 transforms GPλ
to the classical Schur P-function Pλ.
Proposition 4.11**.**
If λ is a strict partition with r parts then
[TABLE]
where we set x⊕y:=x+y+βxy as in (4.6).
Proof.
As in (4.6), set x⊖y:=1+βyx−y.
Fix a strict partition λ with r parts.
Ikeda and Naruse’s first definition of GPλ (see [10, Definition 2.1]) is
[TABLE]
We can rewrite this as
[TABLE]
where Sn−r acts on the variables xr+1,xr+2,…,xn.
Lemma 4.7 implies that
1=∑w∈Sn−rw(∏r+1≤i<j≤nxi⊖xj∏i=r+1nxin−i)
since the left side is ∂1r×wn−r(β)G1r×wn−r=G1.
Multiplying the right side of (4.8) by this expression gives
[TABLE]
which is equivalent to
the desired formula
by Lemmas 4.7 and 4.8.
∎
4.4 Grassmannian formulas
We are ready to state the main new results of this section.
Fix z∈I∞FPF.
The symplectic code of z is the sequence of integers
[TABLE]
The symplectic shape λSp(z) of z is the transpose of the partition
sorting cSp(z).
For example,
if n∈2P and z=n⋯321∈I∞FPF
then
[TABLE]
Define y∈I∞
to be the involution with
[TABLE]
This means that y(i)=i if z(i)=i±1.
In the sequel, we set dearc(z)=y.
The operation dearc is easy to understand in terms of the arc diagram
{{i,z(i)}:i∈P} of z∈I∞FPF. The arc diagram of dearc(z) is formed from that of z by deleting each edge {i<j}
with e<z(e) for all i<e<j.
Recall that i is a visible descent of z∈I∞FPF if z(i+1)<min{i,z(i)}
Definition 4.12** ([9, §4]).**
An element z∈I∞FPF is FPF-Grassmannian if
[TABLE]
for a sequence of integers 1≤ϕ1<ϕ2<⋯<ϕr≤n.
In this case, one has
[TABLE]
by [9, Lemma 4.16], and n is the last visible descent
of z.
We allow r=0 in this definition; this corresponds to the FPF-Grassmannian involution Θ∈I∞FPF with dearc(Θ)=1.
For a given strict partition λ with r<n parts,
there is exactly one FPF-Grassmannian involution z∈I∞FPF with shape λSp(z)=λ
and last visible descent n.
Example 4.13**.**
The involution z=47816523=(1,4)(2,7)(3,8)(5,6)∈I∞FPF
is FPF-Grassmannian with dearc(z)=(2,7)(3,8) and λSp(z)=(4,3).
Define
πb↘a(β):=πb−1(β)πb−2(β)⋯πa(β)
for 0<a≤b,
with πi(β) given by (4.3).
Proposition 4.14**.**
Suppose z∈I∞FPF−{Θ} is FPF-Grassmannian with last visible descent n
and shape λSp(z)=(n−ϕ1,n−ϕ2,…,n−ϕr),
so that
[TABLE]
for some integers 1≤ϕ1<ϕ2<⋯<ϕr≤n.
Then
[TABLE]
where xi⊕xj:=xi+xj+βxixj.
We need two lemmas to prove this proposition.
Lemma 4.15**.**
If a≤b then ∂b↘a(β)(xae)=(−β)b−a−e for e∈{0,1,2,…,b−a}.
Proof.
Since ∂i(β)(1)=−β,
it is enough to check that ∂b↘a(β)(xab−a)=1.
As
[TABLE]
we have
∂b↘a(β)(xab−a)=(−βxa)b−a+(1+βxa)∂b↘(a+1)(β)(∂axab−a).
By induction
[TABLE]
so the lemma follows.
∎
Lemma 4.16**.**
If a≤b and sif=f for a<i<b, then
πb↘a(β)(f)=∂b↘a(β)(xab−af).
Proof.
Assume a<b. It holds by induction
that
[TABLE]
Since
∂a(β)(xab−af)=xa+1b−a−1(πa(β)f+βxaf)+xaf⋅∂a(β)(xab−a−1),
we have
[TABLE]
From here, it suffices to show that
β⋅∂b↘(a+1)(β)(xa+1b−a−1)+∂b↘a(β)(xab−a−1)=0
and this is immediate from Lemma 4.15.
∎
Proof of Proposition 4.14.
Setting β=0 recovers [9, Lemma 4.18]; the proof for generic β
is similar.
Let Ψn,r(x)=∏i=1r∏j=i+1nxixi⊕xj.
Then xλSp(z)Ψn,r(x) is symmetric in xr+1,xr+2,…,xn.
For any j∈[r], the expression
[TABLE]
is symmetric in xj,xj+1,…,xϕj
since if i∈{j,j+1,…,ϕj−1}
then either i=ϕj−1 and
πi(β)θj=θj
or
i<ϕj−1 and
[TABLE]
by the braid relations for πi(β) and induction.
Using Theorem 2.5,
we can rewrite
[TABLE]
where w∈I∞FPF is the Sp-dominant involution satisfying dearc(w)=(1,n+1)(2,n+2)⋯(r,n+r).
Hence by
Lemma 4.16 we have
[TABLE]
It is straightforward from Theorem-Definition 2.3 to show that this
is GzSp.
∎
We can now prove the obvious identity suggested by the notation “GPzSp”:
Theorem 4.17**.**
If z∈I∞FPF is FPF-Grassmannian then GPzSp=GPλSp(z).
Proof.
Assume z∈I∞FPF is as in Proposition 4.14. Then
[TABLE]
so
GPzSp=N→∞limπwN(β)GzSp=GPλSp(z)
by Corollary 4.6 and Proposition 4.11.
∎
Let Pstrict denote the set of strict partitions.
Corollary 4.18**.**
If λ∈Pstrict then
GPλ∈N[β]-span{Gμ:μ∈P}.
Proof.
If λ∈Pstrict then there is an FPF-Grassmannian
z∈I∞FPF with λSp(z)=λ,
and [20, Corollary 4.7] shows that GPzSp∈N[β]-span{Gw:w∈S∞}.
The corollary therefore follows from
Theorems 4.4 and 4.17.
∎
Remark 4.19**.**
As noted in the introduction, one can also derive this corollary from [5, 22].
One needs to compare
[5, Theorems 1.4 and 2.2 and Proposition 3.5] with
[22, Lemma 3.2 and Theorems 3.16, 6.11, and 6.24].
There is a “stable” version of the transition equation for GzSp.
Let SZ denote the group of permutations of Z with finite support.
Write ΘZ for the permutation of Z with
i↦i−(−1)i and let
[TABLE]
Define ℓFPF(z) for z∈IFPFZ by modifying the formula (3.2)
to count pairs (i,j)∈Z×Z;
then ℓFPF(ΘZ)=0 and
(3.3) still holds.
We again write y⋖Fz for y,z∈IFPFZ if ℓFPF(z)=ℓFPF(y)+1
and z=tyt for a transposition t∈SZ.
Identify I∞FPF with the subset of z∈IFPFZ
with z(i)=ΘZ(i) for all i≤0.
Let σ:Z→Z be the map i↦i+2.
Conjugation by σ preserves IFPFZ,
and every z∈IFPFZ
has σnzσ−n∈I∞FPF for all sufficiently large n∈N.
We define
[TABLE]
Also let GPzSpuij:=GP(i,j)z(i,j)Sp for i<j and extend by linearity.
In this context, uij is a formal symbolic operator, not a well-defined linear map.
Corollary 4.20**.**
Fix v∈IFPFZ and j,k∈Z with v(k)=j<k=v(j).
Suppose
[TABLE]
are the integers such that v⋖F(i,j)v(i,j) and v⋖F(k,l)v(k,l).
Then
[TABLE]
Proof.
Define Asc−(v,j,k)={i1,i2,…,ip} and Asc+(v,j,k)={l1,l2,…,lq}.
If m∈N is sufficiently large
then Asc±(Θ2m×v,2m+j,2m+k)=2m+Asc±(v,j,k),
so we obtain this result by taking the limit of Theorem 3.8.
∎
The preceding corollary is a K-theoretic generalization of [9, Theorem 3.6].
The latter result has an “orthogonal” variant given by [8, Theorem 3.2].
Corollary 4.21**.**
Let k∈P be the last visible descent of z∈IFPFZ.
Define v∈IFPFZ as in Corollary 3.11
and
let I={i1<i2<⋯<ip} be the (possibly nonpositive) integers with i<j:=v(k) and v⋖F(i,j)v(i,j).
Then
[TABLE]
where if A={a1<a2<⋯<aq}⊂I then uAj:=ua1jua2j⋯uaqj.
Proof.
The proof is the same as for Corollary 3.11,
now using Corollary 4.20. ∎
This gives a positive recurrence for GPzSp.
We expect that one could use this recurrence
and the inductive strategy in [1, 9, 16]
to prove the following
theorem.
However, a direct bijective proof is already available in [19]:
Theorem 4.22** ([19, Theorem 1.9]).**
If z∈I∞FPF then
[TABLE]
Combining Theorems 4.17 and 4.22
gives this corollary:
Corollary 4.23**.**
If z∈I∞FPF then
[TABLE]
