Irreducible calibrated representations of periplectic Brauer algebras and hook representations of the symmetric group
Mee Seong Im, Emily Norton
TL;DR
This paper constructs an infinite sequence of irreducible calibrated representations of periplectic Brauer algebras, linking them to hook representations of symmetric groups and providing a new categorification of Pascal's triangle.
Contribution
It introduces a novel tower of representations using the degenerate affine periplectic Brauer algebra, expanding understanding of non-semisimple categorifications.
Findings
Formulas for matrices in Jucys--Murphy eigenbasis
Complete spectrum description of the representations
New categorification of Pascal's triangle
Abstract
We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar to that of the degenerate affine Hecke algebra in representation theory of the symmetric group. We write formulas for the representing matrices in the basis of Jucys--Murphy eigenvectors and we completely describe the spectrum of these representations. The tower formed by these representations provides a new, non-semisimple categorification of Pascal's triangle. Along the way, we also prove some basic results about calibrated representations of the degenerate affine periplectic Brauer algebra.
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Irreducible calibrated representations of periplectic Brauer algebras and hook representations of the symmetric group
Mee Seong Im
Department of Mathematical Sciences, United States Military Academy, West Point, NY 10996 USA
and
Emily Norton
Max Planck Institute for Mathematics, 53111 Bonn Germany
Abstract.
We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar to that of the degenerate affine Hecke algebra in representation theory of the symmetric group. We write formulas for the representing matrices in the basis of Jucys–Murphy eigenvectors and we completely describe the spectrum of these representations. The tower formed by these representations provides a new, non-semisimple categorification of Pascal’s triangle. Along the way, we also prove some basic results about calibrated representations of the degenerate affine periplectic Brauer algebra.
Introduction
The periplectic Brauer algebra is an example of a finite-dimensional algebra with a “local system of generators,” in the terminology of Okounkov and Vershik [27]. It is a diagram algebra of crossing lines and curving arcs including the usual generators and relations of the symmetric group , echoed by an additional set of generators and relations which produce a signed Temperley-Lieb algebra with loops evaluated at [math]. Together, the signed Temperley–Lieb algebra and the symmetric group generate the periplectic Brauer algebra , according to additional rules governing the interaction between the two algebras and in . The result is a unique and exceptional algebra: if we impose any other choice of value for the loops, then the algebra cannot exist, and this sets it apart from the usual Brauer algebra which can be defined for any loop-value .
The representation theory of the periplectic Brauer algebra is like a partner dance between and with exaggerated binary gender roles. The symmetric group leads and the algebra of curves follows, embellishing and adding complexity. Cell modules are labeled not just by partitions of , but by layer upon layer of partitions – partitions of , , and all of these cell modules have an irreducible quotient except for the cell module labeling the empty partition of [math] when is even [7]. The filtration of a cell module labeled by a partition contains an irreducible representation labeled by in its Jordan–Hölder filtration exactly once if and only if the Young diagram of fits inside the Young diagram of and their difference belongs to a certain set of skew Young diagrams, and this completely describes the composition series of the cell modules [8, Theorem 1]. Many open questions remain, however, such as determining the branching graph of the irreducible representations.
A famous paper by Okounkov and Vershik [27] reinvents the representation theory of the symmetric group starting from the tower of algebras and the commutative subalgebra of generated by the Jucys–Murphy elements . It is interesting to consider some analogues of this theory for periplectic Brauer algebras. By adding a strand on the right again and again, there is a chain of inclusions going on forever, making a tower of algebras. Each algebra has a large commutative subalgebra generated by the periplectic Jucys–Murphy elements defined in [7]. The Jucys–Murphy elements of naturally generalize the Jucys–Murphy elements of . Recall that the Jucys–Murphy elements of the symmetric group are defined recursively as:
[TABLE]
The Jucys–Murphy elements of are defined recursively as:
[TABLE]
Just as commutes with , commutes with under the embedding of the latter as above [7].
A major difference between and is that is not a semisimple algebra [24],[7]. In contrast to the situation for , there is no reason an arbitrary irreducible representation of should possess a basis of joint eigenvectors for the Jucys–Murphy elements . It is an interesting question to determine when this happens. In this paper, we build a natural series of irreducible representations of which have a basis on which acts as a diagonal matrix for all ; we call representations with such a basis calibrated. As an example of an irreducible calibrated -representation, one can always take for a partition of , because -mod embeds in -mod via , but this is not an interesting example since the representation theory of the symmetric group is known. For a fixed , our family of irreducible calibrated representations cuts transversely across the filtration layers of -mod. The calibrated representations we find are very natural: as -representations they are the exterior powers of the standard representation of , so are irreducible already as -representations. The matrices by which the generators act are very similar to the matrices by which the act – they are obtained by removing the diagonal entries and changing some signs. And there are simple recursive formulas for the eigenvalues of the Jucys–Murphy elements on these representations; the eigenvalues of are obtained from the eigenvalues of by adding or subtracting .
In order to construct interesting calibrated representations of , we study the degenerate affine periplectic Brauer algebra defined in [1],[5]. This is the algebra generated by and commuting formal variables satisfying the same relations with the generators and of as the Jucys–Murphy elements . The definition and use of this algebra follow from the same philosophy that leads to the definition and use of the degenerate affine Hecke algebra in the representation theory of as in the work of Okounkov and Vershik [27] and Kleshchev [20].
We now summarize the contents of the paper and our results. In Section 1, we introduce the algebras and and discuss how they are related to each other as well as to more familiar algebras such as and the degenerate affine Hecke algebra, and what consequences this has for their internal structure and their representation theory. We explain how inherits a filtration by “cup-cap ideals” from , we describe the Jacobson radical of and put some upper and lower bounds on the Jacobson radical of . In Section 2 we study calibrated representations of and . We prove that the center of acts trivially on any calibrated -representation, that is, viewed as a symmetric polynomial an element of acts by its constant term (Theorem 24, Corollary 26). Using some basic results about calibrated -representations, which was the subject of a previous paper by the authors and other collaborators [11], we analyze the eigenvalues of on calibrated -representations and show that the eigenvalues of are integers so long as the eigenvalues of are integers (Theorem 28); as a corollary, the eigenvalues of on any calibrated -representation are always integers (Corollary 30). Furthermore, we work out conditions under which we can write a simple formula for these eigenvalues (Lemma 31); this formula is used to study the family of representations constructed in Section 3.
In Section 3 we tackle the construction of a family of irreducible calibrated -representations endowed with a nonzero action of for any . The starting point is Theorem 34. With the right choice of representing matrices for , the -dimensional standard representation of extends to a one-parameter family of calibrated representations of factoring through when the parameter is set equal to [math]. This gives us the first explicit example of an irreducible calibrated representation of or for any which has a nonzero action of the ’s. Once we have the representation in hand, we compute the exterior powers as -representations with respect to the basis of -eigenvectors, and then extend these to calibrated representations of by writing formulas for the actions of and similar to those used for . We prove in Theorem 38 that this does indeed define a representation of , and thus of when . Since is already irreducible as a -representation, is irreducible as an - or -representation.
The results of Section 3 considered for all give us an infinite series of irreducible calibrated representations of the tower of algebras under the sequence of embeddings adding a vertical strand on the right. A natural question is how these representations are related by restriction. In Section 4 42, we prove that our collection of irreducible calibrated representations is closed under taking composition factors of restriction from to , . Theorem 42 states that the restriction of from to is always indecomposable with two irreducible composition factors if : the head and the socle . As for the representations and , they are just the trivial and the sign representation of , respectively, with acting by [math].
The Bratteli diagram (i.e. branching graph) of thus yields a new categorification of Pascal’s triangle with a non-semisimple rule for the arrows in the interior of the triangle. This is not the first time Pascal’s triangle has been categorified by a branching rule in a module category: other notable instances of Pascal categorification include the category of modules over planar rook algebras (a semisimple categorification) [16] and the branching rule for the standard modules over the blob algebra [18] (furthermore, the irreducible quotients of these blob modules have bases given by paths in Pascal’s triangle that avoid certain hyperplanes, see [4] and take the case and ). Our Bratteli diagram also gives a combinatorial rule for computing the eigenvalues of the Jucys–Murphy elements on any irreducible calibrated -representation of the form . We substitute one-row and one-column partitions for the representations forming the vertices of the Bratteli diagram, and then consider all directed paths from the source vertex to a vertex . The content sequence determined by the unique removable box of each partition along such a path gives an element of the spectrum of , where is determined from and by an easy formula. See Section 4.
1. The finite and degenerate affine periplectic Brauer algebras
1.1. Definitions of the algebras
The degenerate affine Hecke algebra, also called the graded Hecke algebra, has an easy presentation by generators and relations.
Definition 1**.**
[12],[23] The degenerate affine Hecke algebra is the -algebra generated by for and by for with relations:
[TABLE]
We see the group algebra of defined in the first three relations. The last relation echoes the recursive definition of the Jucys–Murphy elements of , but since is a free variable, in these have become abstract entities which generate a polynomial algebra. The idea behind the degenerate affine periplectic Brauer algebra is similar and has a Brauer algebra precursor in [26].
Definition 2**.**
[5, Def. 3.1], [1, Def. 39] The degenerate affine periplectic Brauer algebra is the -algebra generated by and for and by for subject to the relations:
[TABLE]
Remark 3**.**
Relation () can be conveniently packaged as the single relation
[TABLE]
emphasizing the origin of the ’s as Jucys–Murphy elements – multiply each side of the equation by to recover (). The algebra is generated by and , , together with .
Remark 4**.**
Our sign conventions follow [1, Def. 39] and differ in some relations from those in [5, Def. 3.1],[24],[7]. We have omitted the relation which was proved in ([1],[5]) to follow from the other relations. The list of relations above is still not minimal [1, Rem. 40]. For example, (ii) follows by multiplying (iii) on the right by then simplifying using (ii). Similarly, (iv) follows from (v) and (i).
The algebra has several important subalgebras. First, there is the polynomial subalgebra . Second, there is the periplectic Brauer algebra.
Definition 5**.**
The periplectic Brauer algebra is the subalgebra of generated by and for .
This algebra was first introduced by Moon to study Schur–Weyl duality for the periplectic Lie superalgebra [24] and has recently been the focus of much attention [22],[7],[8],[9]. The traditional way to work with such algebras is as diagram algebras: is a crossing, is a cup-cap, and the elements of are linear combinations of Brauer diagrams. A Brauer diagram is a pairing of points, of them equally spaced on a horizontal line of height [math] and of them equally spaced directly above on a horizontal line of height . If two points are paired, they are drawn with a line or strand connecting them. We do not draw the points in our diagrams but only the strands. The strands which connect a point at the bottom of the diagram with a point at the top of the diagram are called through strands. The strands connecting points on the top horizontal line are called cups; on the bottom horizontal line, caps.
Example 6**.**
[TABLE]
Multiplication of elements corresponds to stacking the diagram of on top of the diagram of . Any closed loop occurring in a diagram makes the whole diagram [math]:
[TABLE]
The elements generate the subalgebra , while the elements together with generate a copy of an algebra we denote by , which can be thought of as a signed or super version of a Temperley–Lieb algebra, again with any closed loop evaluated at [math]. Note that the subalgebra generated by is not unital; we throw in to make a unital subalgebra. Together the algebras and also generate a subalgebra of . However, and together generate all of as every , see Remark 3.
[TABLE]
An intuitive way to think about is as the diagram algebra generated by the diagram algebra together with where is a bead in position at the top or bottom of the diagram, subject to some local relations for moving a bead across a cup or cap, or through a crossing:
[TABLE]
and such that beads that are “far away” (somewhere off to the left or right) commute with a crossing or a cup-cap, and the beads commute with each other. It is not at all obvious that such an algebra has the basis one would like: a basis of Brauer diagrams with some number of beads (corresponding to ’s) on each string, pushed to an agreed-upon end of the string in a neat and orderly fashion. Such a beaded Brauer diagram is called a normal diagram. Rather than belabor the precise definition, we draw a picture of a normal diagram in :
[TABLE]
An important first result about the algebra is therefore:
Theorem 7**.**
[5],[1] The algebra has a -basis consisting of all normal diagrams.
This can be thought of as an analogue of classical PBW theorems for universal enveloping algebras and flat deformations of skew group rings 111The proof in [5] is a signed version of the proof given by Nazarov for the degenerate affine Brauer algebra [26]. The proof in [1] proceeds by constructing an explicit faithful representation of using the intrinsic connection of with periplectic Lie superalgebras [1, Theorem 2]..
1.2. Some relations in
In addition to the formulas written in [1, Lemmas 4-9] there are nice formulas for disentangling a bunch of beads stuck between a crossing and a cup or cap. The proofs of the formulas follow from easy induction arguments and the formulas in [1, Lemmas 4-9].
Lemma 8**.**
Let . The following equalities hold in :
[TABLE]
1.3. Basics from representation theory
Write -mod for the category of finite-dimensional -representations and -mod for the category of finite-dimensional -representations. We will only consider finite-dimensional representations in this paper and may often just write “representation” when we mean finite-dimensional representation.
Practically speaking, we will think of the elements of as column vectors in with respect to a fixed basis, and if is a representation then we will think of the linear transformations , , as matrices with respect to that basis. We will abuse notation and define representations by writing the generators directly as matrices, dropping the notation .
The categories -mod and -mod are Krull-Schmidt, that is, any finite-dimensional - or -representation has a unique decomposition as a direct sum of indecomposable representations up to isomorphism and permuting the factors. However, -mod and -mod are not semisimple, that is, not every finite-dimensional - or -representation can be decomposed into a direct sum of irreducible representations. In order to determine when a representation of an algebra is indecomposable, we will use the criterion that is indecomposable if and only if is a local ring. In particular, if then certainly is indecomposable (but not necessarily irreducible).
1.4. Quotient maps and inflated representations
It follows immediately from Definition 2 and Theorem 7 that the degenerate affine Hecke algebra is a quotient of the degenerate affine periplectic Brauer algebra:
Lemma 9**.**
There is a surjective homomorphism of algebras
[TABLE]
given by quotienting by the two-sided ideal .
This allows us to construct a pile of representations of for free by “inflation”:
Definition 10**.**
Let be an -representation. Define an representation by precomposing the -action on with the map . That is, has the same underlying -vector space as with and acting the same on as on , and acting by [math].
In particular, any irreducible representation of gives rise to an irreducible representation of by inflation.
In addition to the surjection , also comes equipped with a surjection to .
Definition 11**.**
The ’th Jucys–Murphy element of is defined inductively by , for .
Lemma 12**.**
[1] There is a surjective algebra homomorphism
[TABLE]
given by , and , the ’th Jucys–Murphy element, for and .
The kernel of is the two-sided ideal . Any irreducible representation of also gives rise by inflation to an irreducible representation of by declaring to act by [math], thus extending the action from to all of by making act by the ’th Jucys–Murphy element .
The algebra not only contains as the subalgebra generated by but also has as a quotient by the two-sided ideal . The irreducible representations of are labeled by partitions of . Using the same method of inflating representations along the quotient map then yields, for every partition of , an irreducible -representation where is the irreducible -representation labeled by . By construction, every acts on by [math]. The module category -mod thus contains -mod as the full subcategory generated by all the for a partition of .
1.5. Filtration by cup-cap ideals
There are filtrations of and by two-sided ideals consisting of diagrams with “at least cup-caps,” equivalently, “at most through strands.” The cup-cap filtration organizes the parametrization of irreducible -representations by partitions of , partitions of , partitions of (except that the unique partition of [math] does not label an irreducible representation) [7]. The irreducibles labeled by partitions of have acting by [math] (as we just saw), the ones labeled by partitions of have acting nontrivially but acting by [math] for , and so on. We expect the cup-cap filtration to play a similar role in the description of irreducible -representations.
If we do not say otherwise, “ideal” will always mean two-sided ideal.
Lemma 13**.**
Consider for an arbitrary . The following statements are true:
- (1)
The ideal is equal to the ideal for any , and moreover, this is true in the subalgebra generated by , and ; 2. (2)
Let mod, then acts by [math] on if and only if acts by [math] on ; 3. (3)
Let mod and suppose is a subspace preserved by . Then either no preserves , or is an -subrepresentation of .
Proof.
- (1)
Let . We have
[TABLE]
and
[TABLE] 2. (2)
This follows from the first part: if the generator of a principal ideal annihilates a module then clearly so does every element in that ideal. 3. (3)
Using relations in Definition 2 shows that for all :
[TABLE]
so for any , for some . It follows that if some preserves an -subrepresentation then all of does.
∎
Consequently, the ideal is a principal ideal and any , , can be taken as its generator. That is to say, we can obtain from by just setting for some .
In fact, the ideal is not only a principal ideal but has an interesting filtration by principal ideals. Set , , , , , where if is even and if is odd.
Lemma 14**.**
- (1)
The ideal consists of all elements of whose diagrams contain at most through strands. It can be generated by any element of the form such that for all . 2. (2)
There is a filtration of given by
[TABLE]
Proof.
The statement is obvious for . For , the relations and of Definition 2 () preserve the number of cup-caps in a diagram, while the relation ( preserves the number of cup-caps in the leading term and has a lower order term with one more cup-cap. Far away things commuting (Definition 2 (),()) obviously preserves the number of cup-caps. Using the relations it is clear as in Lemma 13 that generates the ideal of diagrams with at most cup-caps, and that any other collection of independent cup-caps will also do the job. ∎
1.6. The Jacobson radical
The determination of the Jacobson radical of and is an open problem. In this section we establish some obvious lower and upper bounds on the Jacobson radical of in terms of the ideals introduced in Section 1.5 and we determine .
Recall that if is an associative algebra, the Jacobson radical is defined to be the intersection of all maximal left ideals of . Well-known facts about it include [13]:
Lemma 15**.**
- (1)
Let , then if and only if has a left inverse for all ; 2. (2)
The Jacobson radical is a two-sided ideal of ; 3. (3)
; 4. (4)
is semisimple if and only if and is artinian; 5. (5)
If then ; 6. (6)
If is a surjective homomorphism of algebras then ; 7. (7)
acts by [math] on every irreducible -representation.
Let be the Jacobson radical of and let be the Jacobson radical of . Lemma 12 provides a surjective algebra homomorphism , so by Lemma 15(6) we know that . The algebra is finite-dimensional and non-semisimple [24], so by Lemma 15(4) it holds that [7]. For very small it is possible to calculate by hand: when , is only -dimensional with -basis , and . When , is -dimensional. Moon calculated and found that it is -dimensional; consequently, the maximal semisimple quotient of has dimension For any , Coulembier defined a central element given by a polynomial in the Jucys–Murphy elements and he proved that [7, Equation (6.3)]:
[TABLE]
However, it is unknown if is nonzero for general , see the remark preceding [7, Proposition 6.4.3].
We could not find the next lemma in the literature, so we include a proof.
Lemma 16**.**
Let denote the Jacobson radical of . Then .
Proof.
Let and using the relation , write as a polynomial in with coefficients in . As a polynomial in , has the same degree as it does as a polynomial in . Let be the finite-dimensional quotient of by the two-sided ideal generated by where . For generic , the algebra is semisimple and has basis given by the basis elements of up to degree . Let be the quotient map, then and so for any generic -tuple of complex numbers . Taking to be bigger than the degree of we have , but for a generic -tuple of parameters we have , so . ∎
Note that the algebra , however, still fails to be semisimple as it is not artinian, and its representation theory contains plenty of examples of non-semisimple modules.
Lemma 17**.**
Suppose . Then .
Proof.
Write . We have since . On the other hand, Lemma 8 and [1, Lemma 8] imply that for any element of it holds that . Then the element has left inverse , so is in the Jacobson radical. Therefore it also holds that and the statement is proved. ∎
Lemma 18**.**
Suppose . Then .
Proof.
We know that by Lemma 16. Therefore . However, for all since the element is not invertible: multiply on the left by to get . Similarly, since is not invertible (multiply it on the left by to get [math]). ∎
Lemma 19**.**
Let and suppose is even. Then and .
Proof.
Let . Then if and only if has a left inverse for any . Observe that is a closed loop without any through strands, possibly with complicated self-crossings and a bunch of beads on it. As in Lemma 17, using the relations of Lemma 8 and [1, Lemma 4-9], one can show that . To do this precisely, do downward induction on the number of beads on the loop (i.e. the total number of ’s appearing in the expression), for which the base case is the statement that any closed loop in is [math]. Then do downward induction on the number of crossings. The base case for that is [1, Lemma 8]. If a crossing appears directly below a cap or directly above a cup, use Lemma 8 to eliminate the crossing, then the result is [math] by induction. If the crossing is in between two caps or two cups, move any beads out of the way if necessary across the curve of the cap or cup. These moves produce terms with less beads that are [math] by induction. Then apply [1, Lemma 4(a)] or its upside-down version to switch the position of the crossing and the cup or cap, then apply braid relations if necessary until the crossing can be canceled either using or using [1, Lemma 8]. Either way, the number of crossings has been reduced and the result is [math] by induction. Therefore , and so showing that is a left inverse to for any . It follows from Lemma 15 that . Since , it also holds that . ∎
Remark 20**.**
The reason is required to be even is that if is odd, the corresponding element (consisting of as many cup-caps placed side by side as will fit) is definitely not in the Jacobson radical. To see this, consider . Then , so . In the case , is idempotent so it can’t be in . In the case , is idempotent so it cannot be in or . In either case, this means cannot be in or since the Jacobson radical is an ideal. We illustrate the elements involved in the case :
[TABLE]
and then
[TABLE]
where the tangled strand is just pulled straight (with a sign).
2. Calibrated representations and Jucys–Murphy elements
By Definition 2 the variables commute, and moreover, is a maximal commutative subalgebra of [1]. Similarly, generate a commutative subalgebra of [7]. In parallel with the theory of Jucys–Murphy elements of the symmetric group, commutes with , where is the subalgebra generated by and [7]. Lifting this relation to the affine level, commutes with for each , where denotes the subalgebra of generated by , , and (this follows from Definition 2() and ().
One way to get a grip on some representations of is to study those representations on which act semisimply:
Definition 21**.**
- (1)
A representation of is called calibrated if has a basis with respect to which acts on by a diagonal matrix for all . 2. (2)
A representation of is called calibrated if has a basis with respect to which acts on by a diagonal matrix for all .
Calibrated representations of are a special case of calibrated representations of : a representation of is calibrated if and only is a calibrated -representation on which acts by [math].
Example 22**.**
Every -representation of the form where is an irreducible -representation is calibrated: every acts by [math], by definition, and so for all , acts by , the ’th Jucys–Murphy element of . It is a classical theorem in representation theory of the symmetric group that any irreducible -representation has a basis with respect to which act by diagonal matrices [19],[25]; the foundational work of [27] builds up the representation theory of from scratch using the Jucys–Murphy elements.
Example 23**.**
Every calibrated -representation is a calibrated -representation, again by inflation.
The irreducible calibrated -representations were first classified by Cherednik [6] (and see Kriloff and Ram’s work [21] for the classification of the irreducible calibrated representations of degenerate affine Hecke algebras associated to other finite Coxeter groups). Since they are known, we are interested in studying those irreducible calibrated -representations which do not factor through , so those on which does not act by [math] for all .
2.1. Action of the center of on calibrated representations
Now recall the central element discussed in Section 1.6. Moving up to , consider the similar-looking but - rather than -symmetric element:
[TABLE]
The element belongs to the center of [1, Theorem 53]. The surjection sends to , so by Lemma 15(7) we know that acts by [math] on any irreducible -representation which factors through .
For a representation of , write for the matrix by which the element acts on and let be the matrix entries, and similarly for and .
Theorem 24**.**
Suppose is an indecomposable calibrated representation of with a nonzero action of some , . Then acts by [math] on .
Proof.
Let . Let denote the matrices of those generators acting on in the basis of joint eigenvectors for . We must show that for each , there exist such that .
By Lemma 13, the matrix for all . As in the first part of the proof of [11, Theorem 11], straightforward computations using Definition 2() and () show that implies that and , while implies that and for some . Therefore we would be done if for every , there existed such that the matrix had a nonzero entry in row or column . However, for a general indecomposable representation there is no reason this should hold (see [11] for counterexamples to such an expectation when ), and for the rest of the proof we assume this is not the case.
Order the basis of -eigenvectors as follows: are the basis vectors such that for some , there exist such that , i.e., . We have since every acts by a nonzero matrix, by the remarks above, and furthermore, if then . Supposing that , we then have with the property that for all , . Now the representation is assumed to be indecomposable, but every and every preserve the vector space decomposition . . Therefore some must have a nonzero entry with either (i) and , or (ii) and .
Without loss of generality let’s assume (i) (as for situation (ii), the argument is the same). We will show that this forces for some , contradicting the assumption that are all of the basis vectors for which this happens. Since , there exist some such that . There are three cases: , , and but . Case 1. Assume . Using that if (Definition 2()(iii)) and that , we get and and therefore . Case 2. Suppose and . Then . But implies that , so . Case 3. Suppose and . Then and . Considering the ’th matrix entry of the matrix equation yields that since for any ; similarly, thanks to the equation . Then . This completes the proof. ∎
The center of contains a subalgebra given in terms of and symmetric polynomials in [7, Theorem 6.4.2]:
[TABLE]
Similarly, the center of is described (completely) in terms of and symmetric polynomials in .
Theorem 25**.**
[1, Theorem 53] The center of is the subalgebra
[TABLE]
That is, if and only if for some symmetric polynomial and some .
Combined with Theorem 25, Theorem 24 immediately implies that acts trivially on those indecomposable calibrated representations of not factoring through the degenerate affine Hecke algebra:
Corollary 26**.**
Let . Suppose is an indecomposable calibrated representation of with a nonzero action of some , . Then for all .
2.2. Eigenvalues of calibrated -representations
For any fixed , we have a copy of given by , , , and satisfying the defining relations of Definition 2 in the case , with replaced by . We will call such a copy of an -quadruple. In our previous paper [11], we studied the calibrated representations of . The representations of are fundamental for studying calibrated -representations for , just as the representations of play a fundamental role in Okounkov and Vershik’s study of -representations [27, Section 5], or as -triples appear as the basic building blocks for higher rank groups in Lie theory (see the Jacobson-Morozov Theorem and its applications, [10, Theorem 3.7.1 and Section 3.7]).
We state some basic facts about calibrated -representations which follow easily from the defining relations () and () in Definition 2 and appeared in our paper about indecomposable calibrated representations of [11, Theorem 11]:
Lemma 27**.**
[11] Let be an -quadruple in , and let be a calibrated -representation. The matrices of , , , and in the basis of -eigenvectors satisfy:
- •
for all ;
- •
implies that and ;
- •
the formula holds for all ;
- •
.
These facts are crucial for understanding the - or -eigenvalues of a calibrated - or -representation.
Theorem 28**.**
Let be a calibrated -representation. Suppose the eigenvalues of are in . Then the eigenvalues of are in for all .
Proof.
Assume by induction that has integer eigenvalues for some , so in the basis of eigenvectors for the matrix is a diagonal by matrix with integer entries, . Let , , and suppose first that . Then and so and and therefore . If there are two possibilities. Either (i) , or (ii) . In case (i), implies that . So for all such that column of contains a nonzero off-diagonal entry. Then case (ii) only needs to be dealt with when there exists some with for all . Since , we must have . The formula then gives us that , so again by induction. ∎
In fact, the proof just given tells us more than just that the eigenvalues of are in . It also tells us that if has nonzero entries in all the off-diagonal spots where has nonzero entries (i.e. whenever for ) then each diagonal entry of is obtained from the corresponding diagonal entry of by adding or subtracting (i.e. for all ).
Let us consider in more detail the case that but in a calibrated representation. Let and . If we consider the -vector subspace of spanned by and , then the -quadruple acting on this subspace is the same as acting on this subspace, since acts by [math]. The representations of were analyzed in [27, Section 5]. In matrices we have:
[TABLE]
In particular, , and using the equations coming from the off-diagonal entries in the matrix equation and solving for , it follows that as well. As an -representation, this subspace spanned by and is irreducible. Summarizing, we have:
Theorem 29**.**
Let be a calibrated -representation and let be one of the eigenvectors in the basis for . Then for all .
2.3. Eigenvalues of calibrated -representations
Applying Theorems 28 and 29 to the case , we obtain:
Corollary 30**.**
Let be a calibrated -representation. Then the eigenvalues of are in for all , and their entries as diagonal matrices satisfy for all and for each .
Note that when is a calibrated -representation, and , so the first step of determining produces a diagonal matrix with all ’s and ’s for its diagonal entries. Since has [math]’s on the diagonal, is equal to the diagonal of and so all diagonal entries of are . If we order the basis so that the diagonal entries of come first, then we have:
[TABLE]
In particular, the non-zero, off-diagonal entries of coincide with the non-zero entries of . In general the entries of will not be identical to the off-diagonal entries of , but in the situation that for every , has non-zero entries in all of the spots where has off-diagonal, non-zero entries then the eigenvalues of are easy to determine. This situation will be illustrated by the class of examples in Section 3.
Lemma 31**.**
Let be a calibrated representation of . Suppose that for all and all , if and only if . Then
[TABLE]
and we have the closed formula
[TABLE]
where for each .
Proof.
Lemma 27 implies that only if , so the additional assumption we are imposing is really only the converse. Suppose that if . By Lemma 27, implies that and . Therefore if has nonzero entries in row then it has all [math] entries in column , and if has nonzero entries in column then it has all [math] entries in row . Then the same is true for the off-diagonal entries of by our assumption. Let . Using that , this implies , and therefore for all . We compute the equation for the ’th matrix entry, and using the same property of the rows and columns of and the fact that is [math] on the diagonal, we find:
[TABLE]
The closed formula then follows by induction. ∎
2.4. Restriction of irreducible calibrated representations and non-semisimplicity
One might expect based upon experience with more familiar algebras that an irreducible calibrated -representation would restrict to a direct sum of irreducible calibrated -representations. That is, one can ask the question:
Question 32**.**
Let be an irreducible calibrated representation of and consider its restriction where is the subalgebra of generated by . Clearly is calibrated since act by diagonal matrices on given that all do. Does decompose as a direct sum of irreducible calibrated representations of ?
The whole idea of calibrated representations is that of an inductively defined subalgebra acting semisimply so the answer feels like it should be yes. Indeed, in parts of the literature “calibrated” is used synonymously with “completely splittable,” where the latter term means exactly that the restriction of the representation to any Levi or Young subgroup is semisimple. Even in positive characteristic, calibrated and completely splittable representations of (and hence ) coincide [28, Theorem 2.13]. But for , what is the answer?
Answer 33**.**
No. And the smallest counterexample is -dimensional. Let be the -dimensional representation of given by:
[TABLE]
It is easy to check manually that is irreducible as an -representation, and indecomposable as a calibrated -representation under the action of and . And this is no isolated counterexample, it is the first in an infinite series constructed in the next section: see Theorem 42.
3. Construction of the irreducible calibrated representations of and given by exterior powers of the standard representation of
3.1. The standard representation of
We showed in [11] that must act by [math] on any irreducible calibrated representation of . This is no longer true for when : then has a natural -dimensional irreducible calibrated representation , depending on a parameter , on which acts by a nonzero matrix for all and which is irreducible as an -representation – the underlying -representation is the standard representation . When we then is the inflation from to of some irreducible representation of labeled by , a partition of or or , or so on. The reader might be tempted to guess that ; this is incorrect as for a partition of is inflated from to and so every acts on it by [math].
Here are the matrices for the action of the generators of on the representation when , to give an idea. Then we write the formula for arbitrary below. The ’s act as:
[TABLE]
The ’s act as:
[TABLE]
Then we require that acts by
[TABLE]
for an arbitrary . Since for each , can be expressed in terms of , the ’s and the ’s, the matrices for are now completely determined. If we compute them using the formula we find:
[TABLE]
Now we give the general formula for as a representation for , generalizing the example above. Let be the by matrix with all [math] entries except a in row , column .
Theorem 34**.**
Let be the -dimensional -vector space acted on by the following matrices in :
[TABLE]
Then the matrices satisfy the defining relation of given in Definition 2.
The proof that is an -representation is subsumed by the proof of Theorem 38. A Mathematica code for checking that is an -representation for a given value of is in the Appendix Appendix. For the purposes of setting up Theorem 38, we check:
Lemma 35**.**
is a representation of .
Proof.
We need to check that , , and for .
Write , where is the diagonal part of and is the off-diagonal part of . Then with in position , and for , while and . Then , , , and , so .
Next we check the braid relation. Compute
[TABLE]
where the [math] is in row . Let and in the extremal cases or set and equal to [math] in the formulas. Then
[TABLE]
Finally, the relation if is clear. ∎
Theorem 36**.**
The representation is irreducible considered as a -representation, and therefore is irreducible as an -representation (and as an -representation if ). As a -representation, , the standard representation.
Proof.
Let , . We must show that . It is clear that any vector with a single nonzero entry generates as a -representation, so if contains such a vector then we are done. Let be minimal such that . If then with in the ’st spot, so then and we are done. So we many assume that . Then applying to we get that is in . So if then . Suppose . Applying to we then get that is in so we are done unless . Continuing in this way by applying for in succession, we see that if then . If this is the case, then consider , which then generates . So .
Now to see which irreducible representation of this representation is, we use the character of . We have and . The only irreducible character of with these properties is , the character of the standard representation. ∎
Corollary 37**.**
Let . We then have for some and some irreducible -module . It follows that the Jucys–Murphy elements act on by the diagonal matrices given by Theorem 34 with .
3.2. All irreducible representations of are calibrated
Let . There are three irreducible representations of of dimension bigger than , labeled by the partitions . It turns out that we may extend two of these three irreducible -representations in a unique way to a representation of over such that the ’s act by nonzero matrices and the ’s act diagonally with acting by [math].
is calibrated by Theorem 34. Explicitly, the matrices are given by:
[TABLE]
The -representation is the standard representation and . We may then calculate the matrices for acting on :
[TABLE]
It turns out that using the same kind of formulas as we did for for the ’s and ’s results in a representation of and , that is we take:
[TABLE]
The trick for is to take ’s above the diagonal and ’s below the diagonal in the same positions as the nonzero non-diagonal entries of , and [math]’s everywhere else. To get we add the diagonal entries of and .
It remains to consider the partition . The underlying irreducible -representation is -dimensional. We would be forced by the braid relations, by and , by , and the inductive formula for , to choose matrices for the generators of up to conjugation by:
[TABLE]
However, this is not an -representation because then other relations fail: and , for example. So does not yield an - or -representation with nonzero action of the ’s.
We know from [7] that the irreducible representations of are in bijection with . The irreducible representation for a partition of is given by the irreducible representation of with acting by [math] for all ; the Jucys–Murphy elements then act on these representations by the Jucys–Murphy elements of so they act semisimply. It must therefore hold that . We conclude that act semisimply on every irreducible representation of .
3.3. The exterior powers of the standard representation of
Theorem 34 and the examples of Section 3.2 generalize to all exterior powers of the standard representation of , providing an explicit construction of a family of irreducible calibrated -representations. For , the -representation extends to a calibrated representation of and on which the ’s act by nonzero matrices. Although the underlying -representation is labeled by the partition , the representations constructed in the following theorem do not coincide with the irreducible -representations labeled in the classification given in [7] of irreducible -modules.
Theorem 38**.**
Let be the -dimensional standard representation of and let . Then the irreducible -representation admits a nonzero action by every and a semisimple action by every for and , with acting by the scalar matrix for some , endowing with the structure of an irreducible calibrated -representation which we denote by . When then is an irreducible calibrated -representation .
Proof.
The -representation is irreducible with natural basis given by the wedges where is the basis of used in Theorem 34; the dimension of is . We order the basis elements lexicographically. Write the ’th matrix entry of by instead of so that it is more visible. The representation is the -dimensional -vector space equipped with the following actions of the generators of , given as matrices in the basis .
The matrix for .
The matrix for acting on for has [math]’s except for the following nonzero entries:
[TABLE]
The matrix for .
The matrix for acting on for has [math]’s on the diagonal, the same entries as below the diagonal, and minus the entries of above the diagonal. That is, all the entries of are [math] except for:
[TABLE]
The matrix for .
The matrix for is defined recursively by:
[TABLE]
In closed form, .
We have to check that the matrices defined satisfy the relations of in Definition 2. It is clear that the relations involving hold for if and only if they hold in the case . For the rest of the proof, we therefore consider only the case . In the case , the representation factors through so we really only need to check the relations defining the subalgebra generated by ’s and ’s, and that the formula we have declared for the matrix satisfies , the defining relation of the Jucys-Murphy elements. These are relations 2(),()(i),(ii),()(i),(ii),(),()(i),(iii),(v) by Remark 4,(), and the repackaging of () by Remark 3. By Lemmas 35 and 36, is the standard representation of . Then is an irreducible representation of [17], and we wrote the formula for by computing the action of on using the action of on , so it must hold that , , and if . This puts (), ()(i),(ii) to rest. Moreover, is irreducible if it is well-defined as an -representation, since is an irreducible representation of .
If the matrix has entries in a row then it has [math]’s in column , and vice versa; more precisely, has nonzero entries only in rows involving in a wedge power, and only in columns involving or in a wedge power but not . It follows immediately that () holds.
Write where and is the off-diagonal part of . Then where is the matrix with entries . As in the proof of Lemma 35, and , and , and . Then and , so (i) holds.
Now we check . First, let us compute the nonzero entries of :
[TABLE]
In order to compute we write as the sum of its four types of nonzero entries as above and compute their products with one by one. The diagonal entry of in row multiplies the entries of in row by , which gives us all the nonzero entries of except for those with which occur only for the lower diagonal entries; these are obtained from:
[TABLE]
All other products of off-diagonal entries are [math]. This shows . A similar calculation shows that .
Next, we check relation (iii). We have to show that . As earlier in the proof we compute products with by breaking it into the sum of its diagonal part and off-diagonal part . Computing we have:
[TABLE]
and by a computation similar to that used in the check of we have
[TABLE]
whereas
[TABLE]
Computing , we find that ; while has entries in the same places as in with ’s except in those below-diagonal entries where the wedge labeling the column contains , and thus we obtain that . It follows that . A similar calculation verifies (v).
Next, we check the relations in (i),(ii). Since is an -representation, the relations if are automatically satisfied. The calculation in the proof of Lemma 13(3) shows that for all . Thus for (i) it is enough to check that for all , and for (ii) it is then enough to check that .
First, we check for all . The nonzero entries of are given by:
[TABLE]
The nonzero entries in the product are given by:
[TABLE]
The nonzero entries in the product are given by:
[TABLE]
So for any . By the remarks above, this implies (i) holds.
Let’s check that . The nonzero entries of whose row entry begins with a or whose column entry begins with a are given by:
[TABLE]
However, because in the nonzero entries of , the last two types of term for never match up to with the rows or columns of the nonzero entries of since has no entries in rows or columns labeled . Computing the nonzero entries of and we get for all :
[TABLE]
So . By the remarks above, this implies (ii) holds.
The last thing to check is , as the single relation . Break into the sum of its diagonal and off-diagonal parts: . By a check similar to the check that and commute, and commute for all . Obviously and commute since they are diagonal matrices. So and commute for all . We then have
[TABLE]
and so the relation holds if and only if . Let’s check this. The diagonal matrix has entries given by:
[TABLE]
We have
[TABLE]
and an argument similar to that used to verify also shows . We compute:
[TABLE]
where there is no case distinction for the lower-diagonal term since in that case, . On the other hand,
[TABLE]
where there is no case distinction for the lower-diagonal term, this time because
. The only nonzero terms of are then:
[TABLE]
and on the other hand, the only nonzero entries of are:
[TABLE]
Therefore, we have:
[TABLE]
and so
[TABLE]
as desired.
This finishes the check of the relations giving a calibrated representation of , and thus of a calibrated -representation on which acts by a scalar . ∎
Corollary 39**.**
The eigenvalues of acting on lie in . Moreover, acts by the scalar matrix .
Proof.
The ’th diagonal entry of is the sum of the ’th diagonal entries of for . The diagonal entries of are and . For a given , the diagonal entry corresponding to a wedge is if for some and otherwise. The number of times occurs in the ’th position on the diagonals of all the ’s is therefore equal to , the number of times occurs in the ’th position on the diagonals of all the ’s is equal to . This gives the bounds on the eigenvalues, and since it follows that is a scalar matrix with scalar the total number of ’s minus the total number of ’s which is . ∎
Example 40**.**
Let us continue with the example of when . We saw that the standard representation of extends to a calibrated representation of with nonzero action of for . Let , , , be the basis of simultaneous eigenvectors for , , with respect to which we wrote the matrices in Theorem 34. By Theorem 38, and also extend to irreducible calibrated representations of with nonzero action of . Here we give the explicit matrices for the actions of the generators of by matrices on these two representations.
- (1)
Consider with basis . The matrices for are given by
[TABLE]
The matrices for are given by:
[TABLE]
The matrices for are given by:
[TABLE]
Observe that does not act by [math] and that also does not act by 0. This means that the irreducible is of the form with a partition of [7]. 2. (2)
Consider with basis . The matrices for are given by:
[TABLE]
The matrices for are given by:
[TABLE]
The matrices for are given by:
[TABLE]
Observe that does not act by [math] but does for . This means that the irreducible is of the form with a partition of [7].
4. Pascal’s triangle and a Bratteli diagram
In this section we continue to follow the Okounkov–Vershik approach [27] in order to describe the spectrum and branching of the irreducible calibrated -representations constructed in the previous section. For similar techniques applied to diagram algebras, see for instance the representation theory of the partition algebra as treated by [14],[15],[2]; see [3] for an axiomatic approach to towers of diagram algebras admitting a cellular basis.
Definition 41**.**
If is a calibrated representation, its spectrum is the set of all such that there is a with for .
An element of corresponding to an eigenvector is exactly the sequence of the ’th matrix entries of if is the ’th basis vector for . By considering the branching graph of the calibrated representations , we will see how to easily describe in a completely explicit, combinatorial way.
4.1. Restriction of from to
The underlying -representation of is , which restricts from to as:
[TABLE]
We would like to know how the irreducible calibrated -representation decomposes into irreducible representations when restricted to the copy of generated by and whether the restricted representation is semisimple.
Theorem 42**.**
For , the restriction of the irreducible calibrated -representation is indecomposable, and there is a non-split short exact sequence of calibrated -representations
[TABLE]
For the extremal cases and we have and .
Proof.
The extremal cases of and are obvious since these are the trivial and the sign representations of , respectively. For the rest of the proof we assume that . Any -representation has a unique decomposition as a direct sum of irreducible -representations, and if is irreducible as a -representation then obviously is irreducible as an -representation. So there are two possibilities: either (i) is indecomposable with two distinct composition factors isomorphic to and as -representations; or (ii) is the direct sum of two irreducible -representations, isomorphic to and as -representations. In case (i), we have to further determine whether is irreducible or whether it has two irreducible composition factors.
First, we establish that case (i) holds: is indecomposable for any . To check the cases , we compute and find that it is a local ring. First, we claim that is contained in diagonal matrices. Let us compute the commutant of the matrices for : write for the subalgebra of matrices commuting with . Obviously, . Since not only but also is a multiple of the identity matrix by Corollary 39, we may throw in and at no cost: . Let . Now, suppose that for some , , and some , it holds that and . Computing the ’th matrix entry of we get that it is minus the ’th matrix entry of , but , and therefore . By the same argument, . However, by the definition of the action in Theorem 38 we have whenever and whenever for all . It is clear from the definition of the matrix of in Theorem 38 that for any there is some such that . Thus all off-diagonal entries of vanish: for all , and so is a diagonal matrix. Therefore is contained in diagonal matrices.
Next, we claim that the diagonal matrices commuting with are just the scalar matrices. Observe that are all of the such that has nonzero entries above the diagonal, and that all nonzero entries above the diagonal are ’s. Moreover, if then for , where are wedges of integers from with the wedges ordered lexicographically, i.e. the first row of the matrix is labeled and the last is labeled . If we take the sum then the entries above the diagonal of are [math]’s and ’s. The position of the ’s is defined in terms of increasing some to in the wedge , where . Starting from the first row and then successively increasing one of the numbers in the wedge by one at a time (where can only be increased to if does not already appear in the wedge), one traverses every in a row or column. It follows that the only diagonal matrices commuting with are the scalar matrices. Therefore This proves is indecomposable.
Now that we know is indecomposable, we have to determine whether it is irreducible or an extension of two irreducible representations. Let
[TABLE]
Then . We claim that is an -subrepresentation of isomorphic to and that with basis given by the images of for . First, observe that these vector subspaces indeed have the desired dimensions:
[TABLE]
and thus
[TABLE]
Next, it is clear from the definition of the matrix that preserve the subspace . Indeed: let and consider a wedge labeling a row of with off-diagonal nonzero entries and labeling a basis vector of the vector space complement to in . Then has off-diagonal entries in the columns of this row (so long this does not cause a repeated entry making the wedge equal to [math].) In either case, the column of the nonzero entry contains an in it so does not label one of the basis vectors of . Thus if , , is one of the eigenvectors in then . Moreover, must be irreducible as an -representation since has exactly two composition factors as an -representation and .
Moreover, it is clear that preserves since the only nonzero entries of are given by . Then by Lemma 13(3) it follows that is an irreducible -subrepresentation of . As and has at most two composition factors, it follows that is an irreducible -representation as well. Now, is calibrated, the matrices for acting on are exactly those of , the matrices of have their nonzero entries in exactly the off-diagonal spots where have nonzero entries with s above the diagonal and ’s below the diagonal, and the eigenvalues of on are then given by Lemma 31 and match the formula in Theorem 38. So . By the same argument, . Therefore we have a short exact sequence
[TABLE]
as claimed. ∎
4.2. A Bratteli diagram
Consider the set of nonempty row or column partitions of size congruent to mod and at most :
[TABLE]
Label the representations of , , by the elements of as follows:
Now define a graph whose vertices at level are given by . There is an arrow from at level to at level in if and only if and corresponds to , to . Then is the branching graph of the tower of irreducible calibrated representations we have constructed in this section. The graph is isomorphic to Pascal’s triangle; from level to level it looks like this:
[TABLE]
Reading across the row at level from left to right, the partitions correspond to
Instead of labeling the vertices of the branching graph with partitions, let’s label the vertices with using restriction rule for the arrows. Doing this, we obtain Pascal’s triangle:
[TABLE]
Thus our Bratteli diagram of irreducible calibrated representations can be seen as a categorification of Pascal’s triangle. Pascal’s triangle also shows up in the Bratteli diagrams of the blob algebra [18] and the planar rook algebra [16].
Finally, there is one more natural way to label the vertices of our branching graph. Consider again the Young diagrams of partitions in labeling the vertices of . There is an obvious combinatorial rule to build this graph of Young diagrams: start with the Young diagram consisting of a single box at level of the graph. The Young diagrams at level are built from those at level by adding or removing a single box so long as the result is a non-empty Young diagram with a single row or column. There is an arrow if , , and is obtained from by adding or removing a box. Now, we observe that a single row or column partition has a unique removable box. We then label the vertex occupied by that partition with the content of its removable box. This produces a graph whose vertices are integers, and which is generated by starting with [math] and then has an arrow from at level to at level if and only if .
[TABLE]
This third presentation of the branching graph records the eigenvalues of the Jucys–Murphy elements acting on the basis of -eigenvectors for the vertices .
Fix a level of the branching graph , say . Fix at level of the graph; sticking with , take, say, . Consider a path in the graph from the source vertex at level to . The set of all such paths label the basis of eigenvectors for under the correspondence above. For example, we can take to be the path from the source of the graph to at level . If is another path from the source to the same vertex , we can get from by modifying some number of times by “going the other way around a diamond.” So for example, if then we obtain from by going around the left side of the topmost diamond in the graph through instead of around the right side of the diamond through . On our third, numerical graph, this corresponds to adding to the eigenvalue of .
In general, let be a vertex in . The paths from the source vertex to at level , and thus the spectrum of the irreducible representation -mod, are given by all -tuples of integers of the following form:
[TABLE]
where is the content of the unique removable box of , so
[TABLE]
Starting from any given path from at level to at level , we can obtain any other path with the same starting and ending points by successively adding or subtracting from various , so long as we do not change , we do not change , and at each step of the process we preserve the property that for all . A canonical way of doing this is to start with the vector which is the path to whose first steps are to the left and the remaining steps are to the right. Then generate the rest of the paths by subtracting multiples of from entries of in all possible ways while staying in . In this way we can easily generate for any a vertex in , that is, we can algorithmically generate the matrices which act diagonally on .
Example 43**.**
Let and let . We calculate :
[TABLE]
We can then read off the matrix for , , as the ’th column of this array: , , , and so on. In particular we see that acting on has a simple closed form: it is given by the diagonal matrix whose first entries are and whose last entries are . Symmetrically, there is an easy formula for the matrices of acting on , and we leave this for the reader.
4.3. Final remarks
- We expect that the irreducible module is identified in the labeling conventions of [7] (and possibly up to taking the transpose) as follows:
[TABLE]
That is, we should have:
However, at the time of writing we don’t understand how to make this identification rigorous. Our construction is self-contained and doesn’t use cell modules, whereas the irreducible representation , a partition of or or or is defined as the quotient of the cell module by its radical. The rule for eigenvalues in our Bratteli diagram does not completely match the rule for the eigenvalues of the Jucys–Murphy elements on the cell modules in the Bratteli diagram of up-down tableaux: it differs in the case where a box is removed and the content of the removed box is positive [7, Lemma 6.2.5]. In [7, Lemma 6.2.5] the eigenvalue is always where is the box removed. But in our branching graph of irreducible calibrated representations, the eigenvalue produced by removing a box is if and if . It is worth remarking that our sign conventions for the defining relations of that mix ’s and ’s are opposite to those of [24] and [7]. This has the consequence that when for example, the cell module would have content sequence in our convention, and in the conventions of [7]. We expect that once the differing sign conventions are taken into account, the rest of the discrepancy arises because the cell module -mod, even, is actually irreducible and isomorphic to . The cell module is the unique cell module equal to its radical and does not label an irreducible representation. Perhaps some of the paths we have found factor through at even levels and this accounts for the discrepancy between the rule for cell modules in general and the rule for this particularly special tower of irreducible representations.
- Which irreducible -representations are calibrated? Between the representations constructed in this paper and the representations where is a partition of (the embedding of -mod in -mod), have we found all of them or are there more? To frame the question in terms of where is an irreducible calibrated representation: in this paper we constructed all irreducible -mod such that the -tuples of eigenvalues satisfy for all . Are there irreducible calibrated representations -mod for which is not always but on which doesn’t act by [math]?
Appendix
4.4. Code for Theorem 34
We check that the matrices given in Theorem 34 satisfy the defining relations of in Definition 2 using the code below.
Acknowledgments
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, and the National Security Agency under Grant No. h98230-18-1-0144 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, for two weeks during the Summer of 2018. Lemma 17 was proved during that residence by the first author after conversations with our co-residents in the program, Z. Daugherty and I. Halacheva. The problems of determining the Jacobson radical and calibrated representations of were discussed there with Z. Daugherty and I. Halacheva, and several of their computations of matrix equations arising from a calibrated representation of led to our first paper joint with them [11] and are used here in Lemma 27. We are grateful to Chris Bowman and Kevin Coulembier for helpful communications.
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