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Representation theory of the nonstandard Hecke algebra

The nonstandard Hecke algebra \check{\mathscr{H}}_r was defined by Mulmuley and Sohoni to study the Kronecker problem. We study a quotient \check{\mathscr{H}}_{r,2} of \check{\mathscr{H}}_r, called the nonstandard Temperley-Lieb algebra, which is a subalgebra of the symmetric square of the Temperley-Lieb algebra TL_r. We give a complete description of its irreducible representations. We find that the restriction of an \check{\mathscr{H}}_{r,2}-irreducible to \check{\mathscr{H}}_{r-1,2} is multiplicity-free, and as a consequence, any \check{\mathscr{H}}_{r,2}-irreducible has a seminormal basis that is unique up to a diagonal transformation.


Introduction
Let H r be the type A r−1 Hecke algebra over A = Z[u, u −1 ] and set K := Q(u). The nonstandard Hecke algebraȞ r is the subalgebra of H r ⊗ H r generated by , where C ′ s i and C s i are the simplest lower and upper Kazhdan-Lusztig basis elements, which are proportional to the trivial and sign idempotents of the parabolic sub-Hecke algebra K(H r ) {s i } . The nonstandard Hecke algebra was introduced by Mulmuley and Sohoni in [11] to study the Kronecker problem. The hope was that the inclusion∆ :Ȟ r → H r ⊗H r would quantize the coproduct ∆ : ZS r → ZS r ⊗ ZS r of the group algebra ZS r and canonical basis theory could be applied to obtain formulas for Kronecker coefficients. Unfortunately, this does not work in a straightforward way since the algebraȞ r is almost as big as H r ⊗H r and has A-rank much larger than r!, even though∆ is in a certain sense the quantization of ∆ with image as small as possible (see [2,Remark 11.4]). Nonetheless, as discussed in [2, §1], [10,9], and briefly in this paper, the nonstandard Hecke algebra may still be useful for the Kronecker problem.
Though the nonstandard Hecke algebra has yet to prove its importance for the Kronecker problem, it is an interesting problem in its own right to determine all the irreducible representations of KȞ r . This problem is difficult, but within reach. In this paper, we solve an easier version of this problem.
It is shown in [2] that KȞ r is semisimple. Let τ be the flip involution of H r ⊗ H r given by h 1 ⊗ h 2 → h 2 ⊗ h 1 and let θ : H r → H r be the A-algebra involution defined by θ(T s i ) = −T −1 s i , i ∈ [r − 1]. Twisting an H r -irreducible by θ corresponds to transposing its shape. The algebraȞ r is a subalgebra of (S 2 H r ) θ⊗θ , the subalgebra of H r ⊗ H r fixed by θ ⊗ θ and τ . Based on computations for r ≤ 6, it appears that most of the KȞ r -irreducibles are restrictions of K(S 2 H r ) θ⊗θ -irreducibles, except for the trivial and sign representations of KȞ r .
In this paper we focus on the simpler problem of determining the irreducibles of the nonstandard Temperley-Lieb algebraȞ r,2 , which is a quotient ofȞ r . The algebraȞ r,2 is the subalgebra of H r,2 ⊗ H r,2 generated by , where H r,2 is the Temperley-Lieb algebra (see §5).
The main result of this paper is a complete description of the KȞ r,2 -irreducibles (Theorem 5.1). There are no surprises here: it is fairly easy to show that K(H r,2 ⊗ H r,2 )irreducibles decompose into certain KȞ r,2 -modules. The difficulty is showing that these modules are actually irreducible. We prove this by induction on r and by computing the action of P r−1 on these modules in terms of canonical bases. To carry out these computations, we use results from [5] about projecting the upper and lower canonical bases of a KH r -irreducible M λ onto its KH r−1 -irreducible isotypic components. We also use the well-known fact that the edge weight µ(x, w), x, w ∈ S r , of the S r -graph Γ Sr is equal to 1 whenever x and w differ by a dual Knuth transformation (see §3.2 and §3.5).
One consequence of Theorem 5.1 is that the restriction of a KȞ r,2 -irreducible to KȞ r−1,2 is multiplicity-free. Thus each KȞ r,2 -irreducible has a seminormal basis (in the sense of [12]-see Definition 6.1) that is unique up to a diagonal transformation. This can also be used to define a seminormal basis for any K(H r,2 ⊗ H r,2 )-irreducible. Even though the irreducibles of KȞ r,2 are close to those of K(H r,2 ⊗ H r,2 ), the nonstandard Temperley-Lieb algebra offers something new: the seminormal basis of M λ ⊗ M µ using the chain KȞ J 1 ⊆ · · · ⊆ KȞ J r−1 ⊆ KȞ Jr is significantly different from the seminormal basis using the chain K(H 1,2 ⊗ H 1,2 ) ⊆ · · · ⊆ K(H r−1,2 ⊗ H r−1,2 ) ⊆ K(H r,2 ⊗ H r,2 ), whereȞ J k is the subalgebra ofȞ r,2 generated by P 1 , . . . , P k−1 .
We are interested in these seminormal bases primarily as a tool for constructing a canonical basis of a KȞ r,2 -irreducible that is compatible with its decomposition into irreducibles at u = 1, as described in [2, §19]. Thus even though the representation theory of the nonstandard Hecke algebra alone is not enough to understand Kronecker coefficients, there is hope that the seminormal bases will yield a better understanding of Kronecker coefficients. In fact, [9] gives a conjectural scheme for constructing a canonical basis of a KȞ r,2 -irreducible using its seminormal basis, but this remains conjectural and we do not know how to use it to understand Kronecker coefficients. This paper is organized as follows: sections 2-4 are preparatory: §3 reviews the necessary facts about canonical bases of H r and their behavior under projection onto KȞ r−1irreducibles; §4 gives some basic results about the representation theory ofȞ r . Section 5 contains the statement and proof of the main theorem. Then in §6, seminormal bases of KȞ r,2 -irreducibles are defined, and in §7, the dimension of KȞ r,2 is determined.

Partitions and tableaux
A partition λ of r of length ℓ(λ) = l is a sequence (λ 1 , . . . , λ l ) such that λ 1 ≥ · · · ≥ λ l > 0 and r = l i=1 λ i . The notation λ ⊢ r means that λ is a partition of r. Let P r denote the set of partitions of size r and P ′ r the subset of P r consisting of those partitions that are not a single row or column shape. The symbols , ⊲ will denote dominance order on partitions. The conjugate partition λ ′ of a partition λ is the partition whose diagram is the transpose of the diagram of λ.
The set of standard Young tableaux is denoted SYT and the subset of SYT of shape λ is denoted SYT(λ). Tableaux are drawn in English notation, so that entries increase from north to south along columns and increase from west to east along rows. For a tableau T , sh(T ) denotes the shape of T .
For a word k = k 1 k 2 . . . k r , k i ∈ Z >0 , let P (k), Q(k) denote the insertion and recording tableaux produced by the Robinson-Schensted-Knuth (RSK) algorithm applied to k. The notation Q T denotes the transpose of an SYT Q, so that sh(Q T ) = sh(Q) ′ .
Let T be a tableau of shape λ. If b is a square of the diagram of λ, then T b denotes the entry of T in the square b. If ν ⊆ λ, then T ν denotes the subtableau of T obtained by restricting T to the diagram of ν.
Let λ and µ be partitions of r. Throughout this paper, a 1 , . . . , a k λ (resp. b 1 , . . . , b kµ ) will denote the outer corners of the diagram of λ (resp. µ) labeled so that a i+1 lies to the east of a i (resp. b i+1 lies to the east of b i ), as in the following example.

Canonical bases of the Hecke algebra H r
Here we recall the definition of the Kazhdan-Lusztig basis elements C w and C ′ w and review the connection between cells in type A and tableaux combinatorics, following [5]. We then discuss dual equivalence graphs and recall some results of [5] about projecting canonical bases, which will make these bases fairly easy to work with in the proof of Theorem 5.1.
We work over the ground rings A = Z[u, u −1 ] and K = Q(u). Define K 0 (resp. K ∞ ) to be the subring of K consisting of rational functions with no pole at u = 0 (resp. u = ∞).
Let · be the involution of K determined by u = u −1 ; it restricts to an involution of A. For a nonnegative integer k, the ·-invariant quantum integer is [k] := u k −u −k u−u −1 ∈ A. We also use the notation [k] to denote the set {1, . . . , k}, but these usages should be easy to distinguish from context.
3.1. The Hecke algebra H (W ). Let (W, S) be a Coxeter group with length function ℓ and Bruhat order <. If ℓ(vw) = ℓ(v) + ℓ(w), then vw = v · w is a reduced factorization. The right descent set of w ∈ W is R(w) = {s ∈ S : ws < w}.
For any L ⊆ S, the parabolic subgroup W L is the subgroup of W generated by L.
The Hecke algebra H (W ) of (W, S) is the free A-module with standard basis {T w : w ∈ W } and relations generated by 3.2. The upper and lower canonical basis of H (W ). The bar-involution, ·, of H (W ) is the additive map from H (W ) to itself extending the ·-involution of A and satisfying The A-basis Γ W := {C w : w ∈ W } is the upper canonical basis of H (W ) (we use this language to be consistent with that for crystal bases). Similarly, The A-basis Γ ′ W := {C ′ w : w ∈ W } is the lower canonical basis of H (W ). The coefficients of the lower canonical basis in terms of the standard basis are the Kazhdan-Lusztig polynomials P ′ x,w : (Our P ′ x,w are equal to q (ℓ(x)−ℓ(w))/2 P x,w , where P x,w are the polynomials defined in [7] and . Then the right regular representation in terms of the canonical bases of H r takes the following simple forms: The simplicity and sparsity of this action along with the fact that the right cells of Γ W and Γ ′ W often give rise to C(u) ⊗ A H (W )-irreducibles are among the most amazing and useful properties of canonical bases.
We will make use of the following positivity result due to Kazhdan-Lusztig and Beilinson-Bernstein-Deligne-Gabber (see, for instance, [8]). 3.3. Cells. We define cells in the general setting of modules with basis, as in [5] (this is similar to the notion of cells of Coxeter groups from [7]).
Let H be an R-algebra for some commutative ring R. Let M be a right H-module and Γ an R-basis of M. The preorder ≤ Γ (also denoted ≤ M ) on the vertex set Γ is generated by the relations if there is an h ∈ H such that δ appears with nonzero coefficient in the expansion of γh in the basis Γ.
Equivalence classes of ≤ Γ are the right cells of (M, Γ). The preorder ≤ M induces a partial order on the right cells of M, which is also denoted ≤ M . We say that the right cells Λ and Λ ′ are isomorphic if (RΛ, Λ) and (RΛ ′ , Λ ′ ) are isomorphic as modules with basis. Sometimes we speak of the right cells of M or right cells of Γ if the pair (M, Γ) is clear from context. We also use the terminology right H-cells when we want to make it clear that the algebra H is acting. It is well known that KH r := K ⊗ A H r is semisimple and its irreducibles in bijection with partitions of r; let M λ and M A λ be the KH r -irreducible and Specht module of

Cells and tableaux. Let
. For any KH r -module N and partition λ of r, let p M λ : N → N be the KH r -module projector with image the M λ -isotypic component of N.
We refer to the basis Γ λ of M A λ as the upper canonical basis of M λ and denote it by {C Q : Q ∈ SYT(λ)}, where C Q corresponds to C w for any (every) w ∈ S r with recording tableau Q. Similarly, the basis Γ ′ λ of M A λ is the lower canonical basis of M λ , denoted {C ′ Q : Q ∈ SYT(λ)}, where C ′ Q corresponds to C ′ w for any (every) w ∈ S r with recording tableau Q T . Note that with these labels the action of C s on the upper canonical basis of M λ is similar to (7), with µ(Q ′ , Q) := µ(w ′ , w) for any w ′ , w such that P (w ′ ) = P (w), Q ′ = Q(w ′ ), Q = Q(w), and right descent sets Similarly, the action of C ′ s on {C ′ Q : Q ∈ SYT(λ)} is similar to (6), with µ(Q ′ , Q) := µ(w ′ , w) for any w ′ , w such that P (w ′ ) T = P (w) T , Q ′ = Q(w ′ ) T , Q = Q(w) T , and right descent sets R(C ′ Q ) = {s i : i + 1 is strictly to the east of i in Q}. (10) See Figure 1 for a picture of Γ (3,2) and Γ ′ (3,2) and right descent sets.  Figure 1: The S r -graph on Γ ′ λ and Γ λ . The presence (resp. absence) of an edge means that µ(Q ′ , Q) = µ(Q, Q ′ ) is 1 (resp. 0). 3.5. Dual equivalence graphs. To work with canonical bases in the proof of Theorem 5.1, we make use the notion of dual equivalence graphs 1 from [1]. Given T, T ′ ∈ SYT(λ), we say that T and T ′ are related by a dual Knuth transformation at i if T ′ is obtained from T by swapping the entries i and i + 1 in T or by swapping the entries i − 1 and i in T .
If T and T ′ are related by a dual Knuth transformation at i, then we also say that there is a DKT i -edge between T and T ′ and write Define the dual equivalence graph (DE graph) on SYT(λ) to be the graph with vertex set SYT(λ) and edges given by the DKT i -edges for all i, 2 ≤ i ≤ r − 1.
We will freely use the result from [7] that T T ′ implies µ(T ′ , T ) = µ(T, T ′ ) = 1. Note that this means that the S r -graph on Γ ′ λ (or Γ λ ) contains the underlying simple graph of the DE graph on SYT(λ) (compare Figures 1 and 2).
It is easy to see that (with the help of Figure 3) Here T a i denotes the entry of T in the square a i (see §2). Figure 3: An edge of the DE graph on SYT(λ) as in (11) for i = 1, j = 2.
3.6. Projected canonical bases. Here we recall some results from [5] about projecting the upper and lower canonical bases of M λ onto the KH r−1 -irreducible isotypic components of M λ . These results will make it fairly easy to work with these bases in the proof of Theorem 5.1. For any L ⊆ S, define (C Q ) L to be the projection of C Q onto the irreducible KH Lmodule corresponding to the right cell of Res Here, for a tableau Q and set Z ⊆ Z, Q| Z denotes the subtableau of Q obtained by removing the entries not in Z.
Maintain the notation of (1) for the outer corners of λ. Define a partial order ⊳ r on SYT(λ) by declaring is the subring of K consisting of rational functions with no pole at u = 0 (resp. u = ∞).
The transition matrix expressing the projected basis {(C ′ Q ) J : Q ∈ SYT(λ)} in terms of the lower canonical basis of M λ satisfies the same properties except is upper-unitriangular instead of lower-unitriangular (i.e. To prove the converse, it suffices to show the existence of certain nonzero µ(Q ′ , Q). The DKT i -edges from (11) suffice.
We will also need the following theorem, one of the main results of [5]. See [5,Example 7.5] for an example of this transition matrix. One consequence of this theorem is that Proof. This follows easily from Lemma 3.2.

The nonstandard Hecke algebraȞ r
The nonstandard Hecke algebra was introduced in [11] to study the Kronecker problem. Its role in the Kronecker problem is discussed in [2, §1] and [9]; some of its representation theory is discussed in [2, §11] and [10], including a complete description KȞ 3 and KȞ 4irreducibles; the problem of constructing a canonical basis forȞ r is discussed and [2, §19] and [9]. The main purpose of this paper is to determine the irreducibles of the nonstandard Temperley-Lieb algebra KȞ r,2 , which is a quotient algebra of KȞ r . Here we assemble some basic facts aboutȞ r from [4,11,2] and prove a few new ones. 4.1. Definition ofȞ r . Recall that S is now defined to be {s 1 , . . . , s r−1 }. We repeat the definition ofȞ r from the introduction: The type A nonstandard Hecke algebraȞ r is the subalgebra of H r ⊗ H r generated by the elements We let∆ :Ȟ r ֒→ H r ⊗ H r denote the canonical inclusion, which we think of as a deformation of the coproduct ∆ ZSr : The nonstandard Hecke algebra is also the subalgebra of H r ⊗ H r generated by . For a ring homomorphism K → A, we have the specialization KȞ r := K ⊗ AȞr of the nonstandard Hecke algebra.
The elements P i and Q i satisfy the quadratic relations P 2 i = [2] 2 P i and Q 2 i = [2] 2 Q i , and P i and P i+1 satisfy a nonstandard version of the braid relation (see [4]). For r ≥ 4, the P i satisfy additional relations which seem to be extremely difficult to describe (see [11]).

4.2.
Representation theory of S 2 H r . The representations ofȞ r are related to those of S 2 H r by the fact thatȞ r ⊆ S 2 H r (see, e.g., [2, Proposition 11.6]), so any S 2 H r -module is anȞ r -module by restriction. We recall the description of the KS 2 H r -irreducibles from [2]. These irreducibles are close to those of KȞ r , and even closer to those of KȞ r,2 , which will be described in §5.
First note that we have the following commutativity property for any H r -modules M and M ′ : where the isomorphism is given by the flip Recall from §2 that P r denotes the set of partitions of size r and P ′ r the set of partitions of r that are not a single row or column shape.
Proposition-Definition 4.2 ([2]). Define the following S 2 H r -modules. After tensoring these with K, this is the list of distinct KS 2 H r -irreducibles

4.3.
Contragradients of H r -modules. Any anti-automorphism S of an A-algebra H allows us to define contragradients of H-modules: let ·, · : M ⊗M * → A be the canonical pairing, where M * is the A-module Hom A (M, A). Then the H-module structure on M * is defined by There is an A-algebra automorphism θ : H r → H r defined by θ(T s ) = −T −1 s , s ∈ S. It is not hard to show that θ is an involution and satisfies θ( Let Z * λ be the SYT of shape λ with 1, . . . , λ 1 in the first row, λ 1 + 1, . . . , λ 1 + λ 2 in the second row, etc. For an SYT Q, let ℓ(Q) denote the distance between Q and Z * λ in the DE graph on SYT(λ). It is not hard to show that for any P ∈ SYT(λ), Proof. We first record the following formulae which are immediate from (6), (7), and R(w 0 w) = S\R(w).
As discussed in [4,2], the inclusion∆ :Ȟ r ֒→ H r ⊗ H r is not a good approximation of the coproduct ∆ ZSr , though it is in a certain sense the closest approximation possible. There are a couple ways thatȞ r behaves like a Hopf algebra, one of which is the following.
where these are equalities of maps fromȞ r to H r . Here µ is the multiplication map for H r and η : K → H r is the unit of H r .
Remark 4.5. It is reasonable to suspect that KȞ r is split semisimple, and indeed, our computations are consistent with this being true. In this paper we show that the nonstandard Temperley-Lieb algebra KȞ r,2 is split semisimple by explicitly determining its irreducibles. We are curious if there is a way to show that KȞ r is split semisimple without explicitly determining its irreducibles.
There are one-dimensional trivial and sign representations ofȞ r , which we denote by ǫ + andǫ − :ǫ be the canonical surjection. We then have the followingȞ r -module homomorphismš To see this, note that in general, if M is an H-module and H is a Hopf algebra with counit ǫ, then it follows from the axiom for the antipode that ǫ Moreover, as a consequence of Proposition 4.6 (i) below,ǫ + ⊆M A λ,λ lies inŠ 2M A λ . Then define S ′M A λ := ker(tr) ∩Š 2M A λ . The decomposition (20) yields the decompositioň Proposition 4.6. The maps (18) and (19) as well as the analogous maps forǫ − can be made explicit using upper and lower canonical bases: Proof. The map I of (18) is given by Applying the isomorphism of Proposition 4.3 (iii) then yields (i), except the equality. The equality in (i) follows from the fact that τ • I :ǫ + ֒→ M A λ ⊗ (M A λ ) ⋄ is anȞ r -module homomorphism (sinceȞ r ⊆ S 2 H r ), the multiplicity of Kǫ + inM λ,λ is 1, and Theorem 3.4.
Define the nonstandard Temperley-Lieb algebraȞ r,d to be the subalgebra of H r,d ⊗ H r,d generated by the elements P s := C ′ s ⊗ C ′ s + C s ⊗ C s , s ∈ S. Let P r,2 be the set of partitions of size r with at most two parts and P ′ r,2 be the subset of P r,2 consisting of those partitions that are not a single row or column shape. Define the index setP r,2 for the KȞ r,2 -irreducibles as follows: This section is devoted to a proof of the main result of this paper: Theorem 5.1. The algebra KȞ r,2 is split semisimple and the list of distinct irreducibles is M α := Kǫ + , for α =ǫ + ∈P r,2 .

Decomposition (2) then follows from (21).
We adopt the convention that restrictions fromȞ r toȞ r−1 are considered with respect to the subalgebra ofȞ r generated by Theorem 5.1 will be proved inductively, using the list of KȞ r−1,2 -irreducibles and the fact that the restriction of a KȞ r,2 -irreducible to KȞ r−1,2 is multiplicity-free. Let i∈[k]M i be a multiplicity-free decomposition of a KȞ r -moduleM into KȞ r−1 -irreducibles. Then any KȞ r -submodule ofM is a direct sum of some of theM i . Suppose thať M ′ = i∈IM i , for some I ⊆ [k], is contained in a KȞ r -submodule ofM . We say thatM ′ glues toM j , j / ∈ I, ifM j ⊆M ′ (KȞ r ); this is equivalent top 1M j (x) = 0 for some x ∈M ′ (KȞ r ). Thus if we show thatM 1 glues toM 2 ,M 1 ⊕M 2 glues toM 3 , . . ., i∈[k−1]M i glues toM k , then this proves thatM is a KȞ r -irreducible. Slight variants of this argument will be used in the propositions in the next subsection.

5.2.
Four propositions on the irreducibility of KȞ r -modules. In this subsection we state and prove Propositions 5.3, 5.5, 5.7, and 5.8, which will be used inductively to show that the KȞ r,2 -modules in (1)-(4) of Theorem 5.1 are irreducible.
Proof. We work with the basis Γ ′ λ ⊗ Γ ′ λ ofM λ,µ . It suffices to show thatM λ−a 1 ,µ−b 1 glues toM λ−a i ,µ−b j for (i, j) = (1, 1), which we do as follows: choose T ∈ SYT(λ) and U ∈ SYT(µ) so that (1) T a 1 = r, and if i = 1 then there is an edge T r−1 Such tableaux exist by (11). Then if i = 1 and j = 1, then s r−1 / ∈ R(C ′ T ), s r−1 / ∈ R(C ′ U ) and C ′ T ⊗ C ′ U P r−1 is computed using the last case of (22). The term C ′ T ′ ⊗ C ′ U ′ appears in the sum. The projection lemma (Lemma 3.5) and the fact that shows that the projection of C ′ T ⊗C ′ U P r−1 ontoM λ−a i ,µ−b j is nonzero (the hypotheses of the lemma are satisfied, which depends in a somewhat delicate way on the form of the last case of (22)). Since C ′ T ⊗ C ′ U ∈M λ−a 1 ,µ−b 1 by Corollary 3.3, M λ−a 1 ,µ−b 1 glues toM λ−a i ,µ−b j . If i = 1 or j = 1, these also glue by the same argument, possibly using the second or third case of (22) instead of the fourth.
Given a vector space M, let τ : Let L ν = K 0 Γ ′ ν = K 0 Γ ν be as defined after Theorem 3.4. Let ≤ be a total order on SYT(λ). Then Proof. By Proposition 4.6 (i), where the equivalence is by Theorem 3.4. As S 2 Γ ′ ν is a basis of S 2 M ν , to prove the lemma, it suffices to show that the left-hand side of (27) is not in the span of Proof. First, if k λ = k µ = 1, then λ = (2) and µ = (1, 1), and the result is clear in this case. We will then assume (k, l) = (1, 1) and prove the proposition using the basis Γ ′ λ ⊗Γ ′ λ ; if (k, l) = (1, 1), the proposition can be proved in a similar way 4 using the argument below with a k λ , b kµ in place of a 1 , b 1 and the basis Γ λ ⊗ Γ λ in place of Γ ′ λ ⊗ Γ ′ λ . The KȞ r−1 -irreducibleM λ−a 1 ,µ−b 1 glues toM λ−a i ,µ−b j for (i, j) / ∈ {(k, l), (1, 1)} by the same argument as in the proof of Proposition 5.3.
The assumption λ ⊲ µ implies k ≥ l. Thus k > 1 since we are assuming (k, l) = (1, 1). We will next show that i≤lM λ−a 1 ,µ−b i glues to S ′M ν andΛ 2M ν . We may assume that |SYT(ν)| > 1 because this is equivalent to S ′M ν andΛ 2M ν being nonzero. Thus by (11), we can choose T, T ′ ∈ SYT(λ) and U ∈ SYT(µ) such that (1) T a 1 = r and there is an edge T r−1 Here U ν denotes the subtableau of U obtained by restricting U to ν. The quantity C ′ T ⊗ C ′ U P r−1 is computed using the second or fourth case of (22): if the second case applies, then the projection lemma shows thať if the fourth case applies, then a careful application of the projection lemma shows thať Let x (resp. −x) denote the right-hand side of (28) (resp. (29)). Since ±(C ′ T ′ ) J ⊗ (C ′ U ) J appears in the expression for ±x and T ′ ν = U ν , it follows that the projection of ±x to Λ 2 L ν is nonzero. This uses that The quantities ±x also have nonzero projection onto S ′M ν by Lemma 5.4. Finally, we need that C ′ T ⊗ C ′ U ∈ i≤lM λ−a 1 ,µ−b i , which holds by Corollary 3.3, to conclude that i≤lM λ−a 1 ,µ−b i glues to S ′M ν andΛ 2M ν . It remains to show that Kǫ + ⊆M λ−a k ,µ−b l glues to some other KȞ r−1 -irreducible of Res KȞ r−1M λ,µ . If not, then it follows thatǫ Here, the specialization N| u=1 of an A-module N A is defined to be Q ⊗ A N A , the map A → Q given by u → 1. We are assuming r ≥ 3, soǫ + P 1 = [2] 2ǫ + . But thenǫ + | u=1 is the trivial QS r -module, which is impossible since λ = µ.
Proof. First note that for any H r ⊗ H r -module M, there holds This also shows that (i) holds with 1 + τ in place of 1 − τ . Theň We say that the modules in a list are essentially distinct irreducibles if the nonzero modules in this list are distinct irreducibles.
λ is an irreducible KȞ r -module.
Proof. We work with the basis Λ 2 Γ ′ λ ofΛ 2M λ . Let i > 1 and assumeΛ 2M λ−a 1 is nonzero. We show thatΛ 2M λ−a 1 glues toM λ−a i ,λ−a 1 as follows: given the assumptions, we can choose T, U ∈ SYT(λ) so that (1) T a 1 = r and there is an edge T r−1 is computed using the fourth case of (22). Lemma 5.6 (i) yields the first equality and the projection lemma yields the equivalence in the followinǧ λ−a 1 glues toM λ−a i ,λ−a 1 (i > 1). We next show that 6M λ−a 1 ,λ−a 2 ⊕Λ 2M λ−a 1 glues toΛ 2M λ−a 2 . Since we can assumě Λ 2M λ−a 2 is nonzero, we can choose T, U ∈ SYT(λ) so that (1) T a 1 = r and there is an edge T r−1 is computed using the fourth case of (22). A careful application of the projection lemma shows thať The last line is nonzero because λ−a 1 glues toΛ 2M λ−a 2 . Note that this argument still works ifΛ 2M λ−a 1 = 0. Repeating the arguments of the previous two paragraphs, one shows that 1<i≤k λM λ−a 1 ,λ−a i ⊕ i∈{1,2}Λ 2M λ−a i glues toM λ−a 2 ,λ−a j for j > 2, 1≤i<j≤k λ , i≤2M λ−a i ,λ−a j ⊕ i∈{1,2}Λ Note that S ′M λ does not necessarily have a multiplicity-free decomposition into KȞ r−1irreducibles, but the proof method explained in §5.1 still gives most of the proof. The KȞ r−1 -irreducible Kǫ + may appear with multiplicity more than one, so it is handled separately.
The case i = k λ and j = 1: s r−1 / ∈ R(C T ) and s r−1 / ∈ R(C ′ U ), so xP r−1 is computed using the fourth case of (23). There holds where the first equivalence is by the projection lemma, the second is by Theorem 3.4, and the equivalences are mod uL λ−a i ⊗ L λ−a j . The last line is nonzero because (C ′ T ′ ) J · (C ′ U ′ ) J appears in the left sum, the coefficients µ(A, T )µ(B, U) are nonnegative (Theorem 3.1), and Γ ′ λ−a i ⊗ Γ ′ λ−a j is a K 0 -basis of L λ−a i ⊗ L λ−a j . The case i = k λ , j = 1 (the i = k λ , j = 1 case is similar): xP r−1 is computed using the third or fourth case of (23). A careful application of the projection lemma yields The second line is nonzero because It follows from Proposition 4.6 (ii) thatp 0 Then by Lemma 5.6 (ii) and Hence the left-hand side of (30) being nonzero implies thatM λ−a k λ ,λ−a 1 glues toM λ−a i ,λ−a j .
Fix i ∈ [k λ − 1] and set ν = λ − a i . Now we show that j≤iM λ−a k λ ,λ−a j glues to S ′M ν . Choose T, U ∈ SYT(λ) so that (1) T a k λ = r and there is an edge T r−1 This is possible since we can assume S ′M ν is nonzero, which is equivalent to |SYT(ν)| > 1. Then C T · C ′ U P r−1 is computed using the third or fourth case of (23) with · in place of ⊗. A careful application of the projection lemma yields the first equivalence below The second equivalence is by Theorem 3.4. It follows from Lemma 5.4 and U = T ′ that the second line is nonzero. By an argument similar to that in the previous paragraph, ∈ j≤iM λ−a k λ ,λ−a j , hence j≤iM λ−a k λ ,λ−a j glues to S ′M ν . By an argument similar to the i = 1 case of the previous paragraph,M λ−a k λ ,λ−a 1 glues to To complete the proof, it suffices to show that xH r ⊆ X A ǫ for any x ∈ X A ǫ . If xH r ⊆ X A ǫ , then Res QS r−1 (xH r | u=1 ) is a direct sum of copies of the trivial QS r−1 -module (where N| u=1 of an A-module N A is defined to be Q ⊗ A N A , the map A → Q given by u → 1). It follows that xH r | u=1 is a direct sum of copies of the trivial QS r -module. But this is impossible since there are no copies of the trivial QS r -module in S ′M λ | u=1 .

Completing the proof.
Proof of Theorem 5.1. The proof is by induction on r. Given that the KȞ r−1,2 -irreducibles of the theorem are distinct, it follows from Propositions 5.3, 5.5, 5.7, and 5.8 that the KȞ r,2 -modules in (1)-(4) are irreducible. The list of KȞ r,2 -irreducibles is complete because Res KȞ r,2 K(H r,2 ⊗ H r,2 ) is a faithful KȞ r,2 -module and all the KȞ r,2 -irreducible constituents ofM λ,µ appear in the list. Also, the split semisimplicity of KȞ r,2 follows from the proofs of Propositions 5.3, 5.5, 5.7, and 5.8 since these work just as well over any field extension of K. We now must show that the irreducibles in the list are distinct. For this we apply Proposition 5.2 and refine the cases as follows: (2) Res KȞ r−1,2 S ′M λ ∼ = 1≤i<j≤k λM λ−a i ,λ−a j ⊕ i∈[k λ ] S ′M λ−a i ⊕Kǫ ⊕k λ −1 + , for +λ ∈ P r,2 , λ = (r − 1, 1).
Assuming that r > 3 we will show that this list of irreducibles does not contain repetitions by showing that the irreducibles have distinct restrictions to KȞ r−1,2 . We do this in two steps: The part of the theorem about the decomposition of K(H r,2 ⊗ H r,2 )-modules into KȞ r,2 -irreducibles is immediate from the definitions in §4.4. Remark 5.9. It is possible that this proof would be easier using a Hecke algebra analog of Young's orthogonal basis (see [13]) instead of the lower and upper canonical bases. However, we believe it to be important to understand the action ofȞ r,2 on the lower and upper canonical basis ofM λ,µ anyway. The canonical bases also have the advantage that all computations except the projections onto KȞ r−1,2 -irreducible isotypic components take place over A[ 1 [2] ] rather than K or some extension of K. Moreover, once the results of §3.6 are in place, the only thing we need to know about the S r -graphs Γ ′ λ and Γ λ are the edges corresponding to dual Knuth transformations; it is likely that a proof using a Hecke orthogonal basis would amount to showing the existence of certain dual Knuth transformations in a similar way.

Seminormal bases
We recall the definition of a seminormal basis from [12], observe that KȞ r,2 -irreducibles have seminormal bases, and give combinatorial labels for the elements of these bases. Definition 6.1. Given a chain of semisimple K-algebras K ∼ = H 1 ⊆ H 2 ⊆ · · · ⊆ H r and an H r -module N λ , a seminormal basis of N λ is a K-basis B of N λ compatible with the restrictions in the following sense: there is a partition B = B µ 1 ⊔ · · · ⊔ B µ k such that N λ ∼ = N µ 1 ⊕ · · · ⊕ N µ k as H r−1 -modules, where N µ i = KB µ i . Further, there is a partition of each B µ i that gives rise to a decomposition of N µ i into H r−2 -irreducibles, and so on, all the way down to H 1 .
If the restriction of an H i -irreducible to H i−1 is multiplicity-free for all i, then a seminormal basis of an H r -irreducible is unique up to a diagonal transformation.
A consequence of Theorem 5.1 and Proposition 5.2 is that the restriction of a KȞ r,2irreducible to KȞ r−1,2 is multiplicity-free. Thus each KȞ r,2 -irreducibleM α , α ∈P r,2 , has a seminormal basisŠN α that is unique up to a diagonal transformation. We adopt the convention to take the seminormal basis with respect to the chain KȞ J 1 ⊆ · · · ⊆ KȞ J r−1 ⊆ KȞ Jr , where J i = {s 1 , . . . , s i−1 } andȞ L (for L ⊆ S) is the subalgebra of H r,2 generated by P s , s ∈ L.
We are interested in these seminormal bases primarily as a tool for constructing a canonical basis of a KȞ r,2 -irreducible that is compatible with its decomposition into irreducibles at u = 1, as described in [2, §19]. Even though the irreducibles of KȞ r,2 are close to those of K(H r,2 ⊗ H r,2 ), the seminormal basisŠN λ,µ of M λ ⊗ M µ using the chain KȞ J 1 ⊆ · · · ⊆ KȞ J r−1 ⊆ KȞ Jr is significantly different from the seminormal basis using the chain K(H 1,2 ⊗ H 1,2 ) ⊆ · · · ⊆ K(H r−1,2 ⊗ H r−1,2 ) ⊆ K(H r,2 ⊗ H r,2 ). Thus even though the representation theory of the nonstandard Hecke algebra alone is not enough to understand Kronecker coefficients, there is hope that the seminormal basesŠN λ,µ will yield a better understanding of Kronecker coefficients.