On bases of some simple modules of symmetric groups and Hecke algebras

We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^{\lambda}$, labelled by $e$-restricted partitions $\lambda$ of $n$, for a cyclotomic KLR algebra $R_n^{\Lambda_0}$ over a field of characteristic $p\ge 0$, with mild restrictions on $p$. If all parts of $\lambda$ are at most $2$, we identify a set $\mathsf{DStd}_{e,p}(\lambda)$ of standard $\lambda$-tableaux, which is defined combinatorially and naturally labels a basis of $D^{\lambda}$. In particular, we prove that the $q$-character of $D^{\lambda}$ can be described in terms of $\mathsf{DStd}_{e,p}(\lambda)$. We show that a certain natural approach to constructing a basis of an arbitrary $D^{\lambda}$ does not work in general, giving a counterexample to a conjecture of Mathas.


Introduction
Let K be a field with a Hecke parameter 0 = ξ ∈ K of quantum characteristic e ∈ Z 2 . We consider the Iwahori-Hecke K-algebra H n (ξ). An important special case occurs when ξ = 1 and K has characteristic e, which implies that H n (ξ) = KS n is the group algebra of a symmetric group.
The Specht H n (ξ)-modules S λ H , parameterised by partitions λ of n, play an important role in the representation theory of H n (ξ). In particular, if λ is an e-restricted partition, then S λ H has a simple head D λ , and all simple H n (ξ)-modules occur in this way. The Specht module S λ H has a Murphy basis indexed by the set Std(λ) of all standard λ-tableaux. In this paper, we investigate whether there is a subset of Std(λ) that naturally labels a basis of D λ .
This question can be made much more precise via the language of Khovanov-Lauda-Rouquier (KLR) algebras [18,24], which is used throughout the paper. Given an arbitrary commutative ring O, we consider the cyclotomic KLR O-algebra R Λ 0 n,O of type A (1) e−1 , which has a natural Zgrading, see §2.2. Brundan and Kleshchev [3] and Rouquier [24] proved that R Λ 0 n,K is isomorphic to H n (ξ). Further, Kleshchev, Mathas and Ram [19] constructed a universal Specht R Λ 0 n,Omodule S λ O by explicit generators and relations such that, in particular, S λ K is isomorphic to the H n (ξ)-module S λ H . We denote by D λ K the (simple) head of S λ K if the partition λ is e-restricted and set D λ K := 0 otherwise. The algebra R Λ 0 n,O is equipped with an orthogonal family of idempotents {1 i | i ∈ I n }, where I := Z/eZ. The q-character of a finite-dimensional R Λ 0 n,O -module M is defined by where I n is the free Z[q, q −1 ]-module with basis I n and dim q (1 i M ) ∈ Z[q, q −1 ] is the graded dimension of 1 i M , see §2.1.
Let λ be a partition of n. To each standard tableau t ∈ Std(λ) one attaches its residue sequence i t ∈ I n and degree deg(t) ∈ Z, which are both defined combinatorially, see [5]  i ∈ I n and v t is homogeneous of degree deg(t) for each t. In particular, defining the q-character of any finite set T of standard tableaux by ch q T := t∈T q deg(t) · i t , (1.2) we have ch q S λ K = ch q Std(λ).
(1.3) Therefore, it is reasonable to require that a desired subset of Std(λ) corresponding to a basis of D λ K should have q-character equal to ch q D λ K . Our main results give a combinatorial construction of such a subset of Std(λ) for an arbitrary field K (as above) when λ = (λ 1 , . . . , λ l ) satisfies λ 1 2; we refer to such partitions λ as 2-column partitions. We refer the reader to [14, §3.3] for a further discussion of the problem in general.
In §3.3, we give a combinatorial definition of a subset DStd e (λ) of Std(λ) for every 2-column partition λ. In order to describe DStd e (λ), we represent standard tableaux as paths in a weight space of Dynkin type A 1 and construct a regularisation map reg e on standard tableaux, which plays a key role throughout.
The following theorem shows that, when char K = 0, the set DStd e (λ) labels a basis of D λ K and, moreover, the composition series of S λ K can be lifted to an arbitrary commutative ring O in an explicit way. where µ = (2 x−j , 1 y+2j ) andD µ O 1 denotesD µ O with the grading shifted by 1. Remarkably, the aforementioned construction of DStd e (λ) also leads to a combinatorial description of the q-character of D λ K when K has positive characteristic. Indeed, given a prime p and a 2-column partition λ, define where the sum is over 2-column partitions µ such that D µ K is a composition factor of S λ K , and where we set DStd e,0 (µ) := DStd e (µ). In Section 4, for any 2-column partitions λ, µ, we identify an explicit subset Std e,p,µ (λ) of Std(λ), which may be seen to correspond to the composition factors D µ K in S λ K . More precisely, Std e,p,µ (λ) = ∅ if and only if D µ K is a composition factor of S λ K , and if this is the case, then ch q Std e,p,µ (λ) = q r e,p,λ,µ ch q DStd e,p (µ), (1 The sets Std e,p,µ (λ) are defined in terms of a map reg e,p from the set of 2-column standard tableaux to itself, which generalises the aforementioned regularisation map reg e . In fact, graded decomposition numbers for 2-column partitions can also be described in terms of reg e,p , see Theorem 4.2. The simple module D λ K is self-dual, which implies that ch q D λ K = ch q DStd e,p (λ) is invariant under the involution given by q → q −1 . A combinatorial proof of this fact is given in Remark 4. 24.
The paper is organised as follows. In Section 2, we review cyclotomic KLR algebras, their Specht modules and the connection with representations of Hecke algebras. In Section 3, we associate a path in a weight space of type A k−1 with every standard tableau whose shape is a k-column partition (for k ∈ Z 2 ) and describe the degrees of standard tableaux in the language of paths. We define the aforementioned regularisation map reg e on Std(λ) and the set DStd e (λ) when λ has at most 2 columns.
Section 4 is combinatorial: we prove Theorem 1.2 and the results outlined after the statement of that theorem. The order in which the results are proved is different from the one above. In particular, Theorem 1.2 is obtained as a consequence of the identities (1.4) and (1.5).
In Section 5, we consider homomorphisms between 2-column Specht modules. Using a row removal result from [10], we construct a homomorphism from S µ O to S λ O , where λ and µ are as in Theorem 1.1(iii), and we describe explicitly the kernel and image of this homomorphism, see Theorems 5.6 and 5.14. This leads to a proof of Theorem 1.1. We also construct exact sequences of homomorphisms between 2-column Specht modules, see Corollary 5.17.
Finally, in Section 6, we remove the condition that λ 1 2 and consider a natural approach to extending the definition of the set DStd e (λ) to an arbitrary partition λ of n, based on the structure of S λ Q and its radical rad S λ Q , in the spirit of [14, §3.3]. We give an example showing that in some cases the resulting set DStd e (λ) is 'too big', which yields a counterexample to a conjecture of Mathas [23].
Throughout, given a, b ∈ Z, we write [a, b] := {c ∈ Z | a c b}. If b 0, we often abbreviate a, . . . , a (with b entries) as a b . If X is a collection of elements of an O-module, we denote the O-span of X by X O . The Z-rank of a free Z-module U of finite rank is denoted by dim Z U . If 1 r < n are integers, we set s r := (r, r + 1) to be the corresponding elementary transposition in the symmetric group S n .

KLR algebras and Specht modules
We fix an integer e 2 throughout the paper. We set I = Z/eZ = {0, 1, . . . , e − 1}, abbreviating i + eZ as i (for 0 i < e) when there is no possibility of confusion. For any n ∈ Z 0 , we write I n = I × · · · × I. We define I n to be the free Z[q, q −1 ]-module with basis I n . The symmetric group S n acts on the left on I n by place permutations. An element of I n denoted by i is assumed to be equal to (i 1 , . . . , i n ); we adopt a similar convention for other bold symbols.
2.1. Graded algebras and modules. By a graded module (over any ring) we mean a Zgraded one. If V is a graded module and m ∈ Z, we denote the m-th graded component of where O is a commutative ring. If M = m∈Z M m is a graded A-module then, for any k ∈ Z, we write M k to denote the graded shift of M by k, which has the same structure as M as an A-module and grading given by M k m = M m−k for all m ∈ Z. If M and N are graded A-modules, then Hom A (M, N ) denotes the O-module of Ahomomorphisms from M to N as ungraded modules. Moreover, if M is finitely generated as an A-module, then Hom A (M, N ) is graded by the following rule: given ϕ ∈ Hom A (M, N ) and m ∈ Z, ϕ ∈ Hom A (M, N ) m if and only if ϕ(M k ) ⊆ N k+m for all k ∈ Z. If O is a field, then by a composition factor of a finite-dimensional A-module M we mean a composition factor of M as an ungraded module, unless we explicitly specify otherwise. For . This yields an involution Consider the quiver Γ that has vertex set I, an arrow i ← i + 1 for each i ∈ I and no other arrows. We write i → j and j ← i if there is an arrow from i to j but not from j to i, and we write i j if there are arrows between i and j in both directions (which only happens for e = 2). Further, we write i / − j if i = j and there is no arrow between i and j in either direction. The quiver Γ corresponds to the Cartan matrix C = (c ij ) i,j∈I of the affine type A (1) e−1 , given by Let O be a commutative ring and let n ∈ Z 0 . The KLR algebra R n = R n,O is the O-algebra generated by the elements subject only to the following relations:

4)
y r y s = y s y r , (2.5) for all i ∈ I n and all admissible r, s (see [18,24]). Consider a root system with Cartan matrix C, with simple coroots {β ∨ 0 , . . . , β ∨ e−1 }, see [17]. To each fundamental dominant weight Λ of this root system, one attaches a cyclotomic quotient R Λ n of R n . In this paper, we will only consider the cyclotomic KLR algebra R Λ 0 n , where Λ 0 is a (level 1) weight satisfying Λ 0 , β ∨ i = δ i,0 for all i ∈ I. The algebra R Λ 0 n = R Λ 0 n,O is defined as the quotient of R n by the 2-sided ideal that is generated by the set The algebras R n and R Λ 0 n are both Z-graded with deg(1 i ) = 0, deg(y r ) = 2, deg(ψ r 1 i ) = −c iri r+1 for all i ∈ I n and all admissible r.
We fix a reduced expression for every w ∈ S n , i.e. a decomposition w = s r 1 . . . s rm as a product of elementary transpositions with m as small as possible. Define noting that (in general) ψ w depends on the choice of a reduced expression for w. By definition, the length of w is (w) := m.

Partitions, tableaux and Specht modules.
A partition is a non-increasing sequence λ = (λ 1 , . . . , λ l ) of positive integers. As usual, we write |λ| = j λ j and say that λ is a partition of n := |λ|. The unique partition of 0 will be denoted by ∅. We always set λ r := 0 for all r > l.
We say that λ is e-restricted if λ r − λ r+1 < e for all r ∈ Z >0 . We denote the set of partitions of n by Par(n) and the set of e-restricted partitions of n by RPar e (n). Given k ∈ Z >0 , define Par k (n) := {λ ∈ Par(n) | λ 1 k} and RPar e, k (n) := RPar e (n) ∩ Par k (n).
The dominance partial order on Par(n) is defined as follows: for any λ, µ ∈ Par(n), we set µ λ if r j=1 µ j r j=1 λ j for all r ∈ Z >0 . The Young diagram of λ is the subset λ = {(a, b) | 1 a l, 1 b λ a } of Z >0 × Z >0 . When drawing diagrams, we represent a node (a, b) as the intersection of row a and column b, with the rows numbered from the top down and the columns from left to right.
A standard tableau of size n ∈ Z 0 is an injective map t : {1, . . . , n} → Z >0 × Z >0 such that (i) the image of t is the Young diagram of some partition λ of n; and (ii) the entries of t are increasing along rows and down columns, i.e. whenever (a, b), (c, d) ∈ λ are such that a c and b d, In this situation, we refer to t as a standard tableau of shape λ and write λ = Shape(t). If 0 m n, we denote by t↓ m the restriction of t to {1, . . . , m}. The set of all standard tableaux of shape λ is denoted by Std(λ). For any k ∈ Z >0 , we set Std k (n) := λ∈Par k (n) Std(λ). If t, s ∈ Std(λ), we write t s and say that t dominates s if Shape(t ↓ m ) Shape(s ↓ m ) for all 0 m n. We define t λ as the standard λ-tableau obtained by filling each row successively, going from the top down, so that t λ (λ 1 + · · · + λ a−1 + b) = (a, b) for all (a, b) ∈ λ . Similarly, t λ ∈ Std(λ) is obtained by successively filling each column, going from left to right, so t λ (λ 1 + · · · + λ b−1 + a) = (a, b), where λ j := #{r ∈ [1, l] | λ r j} for all j ∈ Z >0 .
The symmetric group S n acts on the set of all bijections t : {1, . . . , n} → λ as follows: (gt)(r) = t(g −1 r) for all g ∈ S n and 1 r n. For every t ∈ Std(λ), let d(t) ∈ S n be the unique element such that d(t)t λ = t.
By a column tableau of size n we mean an injective map t : {1, . . . , n} → Z >0 × Z >0 such that, whenever (a, b) ∈ t({1, . . . , n}) and a > 1, we have (a − 1, b) ∈ t({1, . . . , n}) and t −1 (a − 1, b) < t −1 (a, b). (That is, in particular, t is required to increase down columns.) For any k ∈ Z >0 , we denote by CT k (n) the set of column tableaux t of size n such that the image of t is contained in Z >0 × {1, . . . , k} (i.e. the entries of t all belong to the first k columns). Note that Std k (n) ⊆ CT k (n).
The residue of a node (a, b) ∈ Z >0 × Z >0 is defined as res(a, b) = b − a + eZ ∈ I. We refer to a node of residue i as an i-node. The residue sequence of a column tableau t is i t := res t(1) , . . . , res t(n) ∈ I n .
The set of all standard λ-tableaux with a given residue sequence i ∈ I n is denoted by Std(λ, i).
For each standard tableau t ∈ Std(λ), the degree deg e (t) of t is defined in [5] as follows. A node (a, b) ∈ Z >0 is said to be addable for λ if (a, b) / ∈ λ and λ ∪ {(a, b)} is the Young diagram of a partition. We say that (a, b) is a removable node of λ if (a, b) ∈ λ and λ \ {(a, b)} is the Young diagram of a partition. A node (a, b) is said to be below a node (a , b ) if a > a . If (a, b) is a removable i-node of λ, define Finally, we define the degree of the unique ∅-tableau to be 0 and define recursively deg e (t) := d t(n) (λ) + deg e (t↓ n−1 ) for t ∈ Std(λ).
If t ∈ Std(λ) and 1 r s n, we write r → t s if t(r) and t(s) are in the same row of λ . We also write i λ := i t λ .
Let O be a commutative ring. We refer the reader to [19,Section 5] for the definition of a Garnir node A ∈ λ and the corresponding Garnir element g A ∈ R n = R n,O . The universal row Specht module S λ = S λ O is defined in [19] as the left R n -module generated by a single generator v λ subject only to the relations 14) for all i ∈ I n and all admissible r ∈ {1, . . . , n}. By [19,Corollary 6.26], the action of R n on S λ factors through R Λ 0 n , so S λ is naturally an R Λ 0 n -module. For each t ∈ Std(λ), we set v t := ψ d(t) v λ , noting that in general v t depends on the choice of the reduced expression for d(t) made in (2.12).
Proposition 2.1. [19, Proposition 5.14 and Corollary 6.24]. Let λ ∈ Par(n). The Specht module S λ is free as an O-module, with basis {v t | t ∈ Std(λ)}. Moreover, S λ is a graded R Λ 0 n -module, with each v t homogeneous of degree deg e (t).
Hecke algebras at roots of unity, a cellular basis and simple modules. Let F be a field such that-setting p := char F -we have that p = 0, p = e or p is coprime to e. Let the field K be an extension of F , and assume that ξ ∈ K \ {0} has quantum characteristic e, i.e. e is the smallest positive integer such that 1 + ξ + · · · + ξ e−1 = 0.
The Iwahori-Hecke algebra H n (ξ) is the K-algebra generated by T 1 , . . . , T n−1 subject only to the relations for all admissible r and s. The algebra H n (ξ) is cellular, with cell modules S λ H parameterised by the partitions λ of n; see [22].
By the following fundamental results, much of the modular representation theory of Iwahori-Hecke algebras at roots of unity can be phrased in terms of questions about KLR algebras and their universal row Specht modules. [19,Theorem 6.23]. Let K be a field of characteristic p, and suppose that ξ ∈ K \ {0} has quantum characteristic e, where either p = e or p is coprime to e if p = 0, e. There is an algebra isomorphism θ : H n (ξ) We remark that graded modules over R Λ 0 n,K which can be identified with the modules S λ H were originally constructed in [5] and that the proof of the identification of Specht modules in Theorem 2.3 uses results from [5].
Let λ ∈ Par(n). The cell module corresponding to λ in the graded cellular structure of Theorem 2.4 is isomorphic to S λ F as a graded R Λ 0 n,F -module: this follows from [13, Corollary 5.10] and the proof of [19,Theorem 6.23]. In the sequel, we identify S λ F with the corresponding graded cell module. The cellular structure of Theorem 2.4 yields a symmetric bilinear form ·, · on S λ F . This form is determined by the equation The form ·, · is homogeneous of degree 0 and satisfies xu, v = u, xv for all is a complete set of graded simple R Λ 0 n,Fmodules up to isomorphism and grading shift.
Let M be a finite-dimensional graded R Λ 0 n,F -module. Given µ ∈ RPar e (n), the graded compo-  [20] proved that R Λ 0 n,O is cellular for any commutative ring O, generalising Theorem 2.4. Both of these results hold with Λ 0 replaced by an arbitrary dominant integral weight Λ, as do the aforementioned results from [3][4][5]19].

Standard tableaux and paths in the weight space
In this section we fix an integer k 2. In §3.1-3.2, we attach to each standard tableau t with at most k columns a path π t in a weight space for the Lie algebra sl k . We show that the degree of t can be non-recursively described in terms of interactions of π t with certain hyperplanes in that weight space (Lemma 3.3) and that the residue sequence of a tableau is invariant under certain reflections of the corresponding path (Lemma 3.2). In §3.3, we specialise to the case k = 2, which is the only one used in the rest of the paper, and define a regularisation map on paths, which plays a key role in the sequel.

The affine Weyl group. Let
Let Φ be the root system of the Lie algebra sl k (C) with respect to the Cartan subalgebra h of diagonal matrices in sl k (C), with {α 1 , . . . , α k−1 } being a set of simple roots. We consider the (real) weight space V := RΦ = h * R , where h R is the set of matrices in h with real entries. For each i = 1, . . . , k, let ε i ∈ V be the weight sending a diagonal matrix diag(t 1 , . . . , t k ) ∈ h R to t i . Then ε 1 + · · · + ε k = 0, and we may assume that The group W also acts on the set {1, . . . , k} in the natural way.
There is a faithful action of W aff on V defined by If α ∈ Φ and m ∈ Z, consider the hyperplane We refer to H α,m as an α-wall or simply a wall. For each α ∈ Φ and m ∈ Z there exists (unique) s α,m ∈ W aff such that s α,m acts on V by reflection with respect to H α,m , i.e.
for all v ∈ V . We consider the set which may be viewed as the dominant chamber of the Coxeter complex corresponding to Φ. Using (3.1), we see that Given t ∈ CT k (n), for any 0 a n and 1 j k, set i.e. c a,j (t) is the number of elements of t({1, . . . , a}) in the jth column. Define π t ∈ P n by π t (a) := c a,1 (t)ε 1 + · · · + c a,k (t)ε k (a = 0, . . . , n).
Note that the end-point π t (n) of the path π t depends only on the image of t (i.e. only on the shape of t in the case when t is a standard tableau).
Lemma 3.1. The assignment t → π t is a bijection from CT k (n) onto P n and restricts to a bijection from Std k (n) onto P + n .
Proof. The first assertion of the lemma is clear from the definitions. For the second assertion, let t ∈ CT k (n) and observe that t is a standard tableau if and only if c a,j (t) c a,j+1 (t) for all a = 1, . . . , n and all j = 1, . . . , k − 1. The lemma follows by (3.3).
Let π ∈ P n and suppose that π(a) ∈ H α,m for some α ∈ Φ + and m ∈ Z. We define the path s a α,m · π ∈ P n by setting That is, s a α,m · π is obtained by reflecting a 'tail' of π with respect to H α,m . The residue sequence i t of t ∈ CT k (n) may be described as follows: if 1 a n, then where j ∈ {1, . . . , k} is determined by the condition that t(a) is in the jth column. The following lemma shows that reflecting a tail of a path as above does not change the residue sequence of the corresponding tableau.
For any j ∈ {1, . . . , k}, we have is in the (s α j)th column, and it follows using (3.
Remark 3.4. The correspondence between standard tableaux and paths, as above, is considered in [12, Section 5]. The degree function (3.6) is similar to the one defined in [2, Definition 1.4] in a somewhat different context. Also, consider a path π ∈ P + n such that π(n) does not belong to any wall H α,m and π(a) / ∈ H α,m ∩ H α ,m for any distinct walls H α,m and H α ,m whenever 0 a < n. Then one can associate with π a Bruhat stroll as defined in [8, §2.4], and deg e (π) is precisely the defect of the corresponding Bruhat stroll.
We define a map reg e : P + n → P + n as follows. Given π ∈ P + n , we set reg e (π) = π if π(a) / ∈ H for all a ∈ {0, . . . , n}. Otherwise, let a ∈ {0, . . . , n} be maximal such that π(a) ∈ H and, if m ∈ Z >0 is given by the condition that π(a) ∈ H m , define reg e (π) := π if π(n) me − 1; s a m · π if π(n) < me − 1. Less formally, we consider the last point at which the path π meets a wall H m (if such a point exists) and, if this point is greater than the endpoint of π, we get reg e (π) by reflecting the corresponding 'tail' of π with respect to H m . Further, we set r e (π) := 0 if reg e (π) = π; 1 if reg e (π) = π.
The following is clear from the definitions: We have deg e (reg e (π)) = deg e (π) − r e (π) for all π ∈ P + n . By Lemma 3.1, there is a well-defined map reg e : Std 2 (n) → Std 2 (n) determined by the condition that π reg e (t) = reg e (π t ) for all t ∈ Std 2 (n). We also have a map r e : Std 2 (n) → {0, 1} defined by r e (t) := r e (π t ).
For any λ ∈ Par 2 (n), set We refer to the elements of DStd e (λ) as e-regular standard tableaux. The following is easily seen: If y e − 1, let a ∈ {0, . . . , n} be maximal such that π s (a) = me − 1, and let t be the standard tableau determined by the condition that π t = s a m · π s . Then {s} if s ∈ DStd e (µ) and either y < e or j = 0; {s, t} if s ∈ DStd e (µ), y e and j > 0.

Characters and graded decomposition numbers for 2-column partitions
Let F be a field of characteristic p 0. We assume that p = 0, p = e or p is coprime to e, cf. §2.4. We fix n ∈ Z 0 and use the notation of §3.3 throughout the section. In particular, P + n is a set of paths in a weight space of type A 1 , and there is a bijection Std 2 (n) ∼ −→ P + n given by t → π t , see Lemma 3.1.
4.1. (e, p)-regularisation. We now define the (e, p)-regularisation map reg e,p : P + n → P + n , which is needed to state the main results of this section. If p = 0, then set reg e,p := reg e . If p > 0, then reg e,p is defined recursively, as follows. For all π ∈ P + n : (1) If reg ep z (π) = π for all z ∈ Z 0 , then set reg e,p (π) := π.
Note that the recursion always terminates because any map reg m either fixes a path or increases its end-point. We also have a map reg e,p : Std 2 (n) −→ Std 2 (n) determined by the identity reg e,p (π t ) = π reg e,p (t) for all t ∈ Std 2 (n). Given π ∈ P + n , we have reg e,p (π) = reg ep z h (. . . reg ep z 2 (reg ep z 1 (π)) . . .) (4.1) where, for each r = 1, . . . , h, the integer z r 0 is maximal such that reg ep z r−1 (. . . reg ep z 1 (π) . . .) is not an ep zr -regular path. Note that z 1 > · · · > z h . When p = 0, we use the convention that ep 0 = e; in this case, 0 h 1. We refer to (4.1) as the regularisation equation and to Z = {z 1 , . . . , z h } as the regularisation set of π. If t ∈ Std 2 (n), then the regularisation set of t is defined to be that of π t , and the regularisation equation of t is also defined to be that of π t , with π t replaced by t on both sides.  .
One of the main results of §4 is the following theorem, which gives a combinatorial description of the graded decomposition numbers [S λ : D µ ] q (when λ ∈ Par 2 (n)) in terms of the map reg e,p .
Theorem 4.2. Let λ ∈ Par 2 (n), µ ∈ RPar 2 (n), and suppose that either p = e or p is coprime to e if p = 0, e. If s ∈ DStd e,p (µ), then In particular, the right-hand side does not depend on the choice of s.
As we will see, the sum on the right-hand side always contains at most one non-zero term. In §4.2, we give a description of graded decomposition numbers for 2-column partitions that does not use the map reg e,p and refines a known result on ungraded decomposition numbers, see Theorem 4.10. In §4.3, we use this description to prove Theorem 4.2.

Decomposition numbers. Set
so that D p (q) is the submatrix of the graded decomposition matrix of R Λ 0 n,F corresponding to partitions in Par 2 (n). Let D p = D p (q)| q=1 denote the corresponding ungraded submatrix of the decomposition matrix.
In this subsection, we prove a formula for the entries of D p (q). Recall that for p > 0, there is a unique square matrix A p -known as an adjustment matrix-such that D p = D 0 A p . Similarly, there is a unique graded adjustment matrix A p (q) such that D p (q) = D 0 (q)A p (q), see [22,Theorem 6.35], [4, Theorem 5.17]. Write A p (q) = (a λµ (q)) λ,µ∈RPar e, 2 (n) . The following is a special case of [4, Theorem 5.17]: Theorem 4.3. For every λ ∈ Par 2 (n) and µ ∈ RPar e, 2 (n), we haveā λµ (q) = a λµ (q).
Our strategy is as follows. The ungraded decomposition matrices D p for all p > 0 are known by work of James [15,16] and Donkin [7]; the graded decomposition matrices D 0 (q)-and hence the ungraded decomposition matrices D 0 -are given in [21]. As we see below, all entries in the former matrices are either 0 or 1, so it follows by Theorem 4.3 that A p (q) = A p , i.e. every entry of A p (q) is a constant (Laurent) polynomial. Hence we are able to compute the matrix D p (q) = D 0 (q)A p (q). We begin with some notation. Given a, b ∈ Z 0 and assuming that p > 0, we say that a contains b to base p and write b p a if, writing a = a 0 + a 1 p + a 2 p 2 + · · · and b = b 0 + b 1 p + b 2 p 2 + · · · (with 0 a i , b i < p for all i), for each i 0 either b i = 0 or b i = a i . We write b 0 a whenever a 0 and b = 0. Theorem 4.4. Assume that p > 0 and either p = e or p is coprime to e. Suppose that λ = (2 u , 1 v ) ∈ Par 2 (n) and µ = (2 x , 1 y ) ∈ RPar e, 2 (n), with u x. Then Proof. Assume that (i) is false, and let i be the largest index such that δ i+1 = δ i+1 and a i > 0. Without loss of generality, δ i+1 = 0 and δ i+1 = 1, and hence the difference between the left-hand and the right-hand sides of (4.5) is at least This contradiction proves (i), and (ii) follows immediately.
Definition. Let y ∈ Z 0 . If p > 0, then we define the (e, p)-expansion of y to be the expression y = (a s p s + a s−1 p s−1 + · · · + a 1 p + a 0 )e + l − 1, where 0 l < e, 0 a i < p for all i, and if y e − 1 then a s > 0 (cf. [7, §3.4]).
(4. 6) We note that the last two cases in (4.6) cannot occur simultaneously by Lemma 4.5.
(i) We have Theorem 4.8. Suppose that p = 0. If λ = (2 u , 1 v ) ∈ Par 2 (n) and µ = (2 x , 1 y ) ∈ RPar e, 2 (n) are such that u x then [S λ : D µ ] q = f q e,0 (y, u − x). The notation in [21, Theorem 3.1] is different from that of Theorem 4.8 or Lemma 4.7, but it is not difficult to see that the results are equivalent. Alternatively, a direct proof of Theorem 4.8 can be easily obtained via the fact that the decomposition numbers [S λ F : D µ F ] q are certain parabolic Kazhdan-Lusztig polynomials, see [26] or [12,Theorem 5.3], together with Soergel's algorithm [25] for computing those polynomials.
Proof. In view of Lemma 4.5, the result follows from Lemmas 4.6 and 4.7 together with Theorem 4.8.
Theorem 4.10. Suppose that either p = e or p is coprime to e if p = 0, e. If λ = (2 u , 1 v ) ∈ Par 2 (n) and µ = (2 x , 1 y ) ∈ RPar e, 2 (n) with u x, then [S λ F : D µ F ] q = f q e,p (y, u − x). Proof. If p = 0, then this is Theorem 4.8, so we assume that p > 0. Setting q = 1 in Lemma 4.9, we have D p = D 0Ãp , so thatÃ p = A p is the RPar e, 2 (n) × RPar e, 2 (n)-submatrix of the ungraded adjustment matrix corresponding to 2-column partitions. Since the entries of A p are either 0 or 1, it follows from Theorem 4.3 thatÃ p = A p = A p (q), that is, D p (q) = D 0 (q)Ã p . The result then follows by another application of Lemma 4.9.

Proof of Theorem 4.2.
We fix a partition µ = (2 x , 1 y ) ∈ RPar e, 2 (n) and prove Theorem 4.2 for this fixed µ. Recall the sets Std e,p,µ (λ) for λ ∈ Par 2 (n) defined by (4.2). If p = 0, then the statement of Theorem 4.2 is an immediate consequence of Lemma 3.6, Lemma 4.7 and Theorem 4.8. If p > 0, then we set y = (a s p s + a s−1 p s−1 + · · · + a 1 p + a 0 )e + l − 1 to be the (e, p)-expansion of y.
Let m = a s p s + · · · + a z 1 p z 1 . Then d = me + j − 1 for some 0 by Lemma 3.6. Also, a z 1 > 0, for if a z 1 = 0 then either t / ∈ DStd ep z 1 +1 (λ) or t ∈ DStd ep z 1 (λ), contradicting the hypothesis on the regularisation equation of t. The result follows.
The following lemma is an easy consequence of the definitions in §3.3. (ii) Let t ∈ Std 2 (n). Suppose that s := reg ep b (t) = t and e π t (n) + 1. Then t is e-regular if and only if s is not e-regular.
Lemma 4.14. Let λ = (2 u , 1 v ) ∈ Par 2 (n) be such that S λ F has a composition factor D µ F . (i) If s ∈ DStd e,p (µ) then there exists a unique t ∈ Std(λ) such that reg e,p (t) = s. (ii) All elements of Std e,p,µ (λ) have the same regularisation set.
Proof. If p = 0, then the result holds by Lemma 3.6. So, assume that p > 0.
To prove (i), we argue by induction on h. If h = 0, then λ = µ and there is nothing to prove. So, assume that h > 0. Set and let ν := 2 c , 1 d ∈ Par 2 (n), with c = (n − d)/2. By Lemma 4.6 and Theorem 4.10, D µ F is a composition factor of S ν F , so by the inductive hypothesis, there is a unique r ∈ Std(ν) such that reg e,p (r) = s. Moreover, by Lemmas 4.12 and 4.5, reg e,p (r) = reg ep z h (. . . reg ep z 2 (r) . . . ) is the regularisation equation of r. In particular, reg ep z (r) = r for all z > z 2 . Let m = a s p s−z 1 + a s−1 p s−1−z 1 + · · · + a z 1 .
To prove the uniqueness statement in (i), recall that any t ∈ Std(λ) such that reg e,p (t ) = s has regularisation set Z. By the inductive hypothesis, this implies that reg ep z 1 (t ) = r, and by (4.7) we have t = t.
Lemma 4.15. Let λ = (2 u , 1 v ) be a partition of n such that D µ F is a composition factor of S λ F . If t ∈ Std e,p,µ (λ) then [S λ F : D µ F ] q = q re(t) . Proof. If p = 0, then the result holds by Theorem 4.8 and Lemma 4.7, so we assume that p > 0. Observe that δ 0 = h. By Theorem 4.10 and Lemma 4.6, the result follows.
Proof of Theorem 4.2. By Theorem 4.10 and Lemmas 4.6 and 4.12 (or by Lemma 3.6 and Theorem 4.8 in the case p = 0), if Std e,p,µ (λ) = ∅ then D µ F is a composition factor of S λ F . The converse also holds, by Lemma 4.14(i). Moreover, that lemma asserts that, given s ∈ DStd e,p (µ), there exists a unique t ∈ Std(λ) such that reg e,p (t) = s. Further, by Lemma 4.15, we have [S λ F : D µ F ] q = q re(t) in this case, completing the proof. 4.4. Characters of simple modules. Recall the notation of § §3.2-3.3 applied to a weight space of type A 1 . Let u r s v be integers and π ∈ P + [u,v] . Denote the restriction of π to [r, s] by π[r, s], so that π[r, s] ∈ P + [r,s] . We refer to every such restriction π[r, s] as a segment of π. The following is clear: . Definition. Let r < s be integers. Let η ∈ P + [r,s] be such that η(r) = η(s) = me − 1 for some m ∈ Z >0 , so that η(r) ∈ H m .
• If (m − 1)e − 1 < η(c) < me − 1 whenever r < c < s, then we call η a negative arc. If one of the above two statements holds, we say that η is an arc.
Let m ∈ Z >0 and π ∈ P + [u,v] . Recall the reflections s m = s α,m from (3.2). We define s m · π ∈ P [u,v] to be the path obtained by reflecting π with respect to the wall H m , so that (s m · π)(a) = s m · (π(a)) (u a v).
The following result describes the degree of a path in terms of the number of its positive and negative arcs.
Our next goal is to describe the effect of the map reg e,p on the degree of a standard tableau. If p > 0, then given η ∈ P + n and a finite subset Lemma 4.20. Assume that p > 0. Let π ∈ P + n and η = reg e,p (π). Suppose that Z is the regularisation set of π and w(Z, η) = (w 1 , . . . , w h ). Let w 0 = 0 and w h+1 ∈ [0, n] be maximal such that π(w h+1 ) ∈ H, with w h+1 = 0 if no such number exists. Then for all 0 i h.
Proof. When p = 0, the result follows from Lemma 3.5, so we assume that p > 0. Let s = reg e,p (t) and Z be the regularisation set of t, so that reg e,p,λ,µ (t) = ρ Z (s). Write w(Z, π t ) = (w 1 , . . . , w h ). Let w h+1 ∈ [0, n] be maximal such that π t (w h+1 ) ∈ H, with w h+1 = 0 if no such number exists. By Lemma 4.18(ii), for any 0 i h we have Hence, where the second equality is due to Lemma 4.20 and we have used Lemma 4.16 for the first and third equalities.
(i) The element v t ∈ S λ O does not depend on the choice of a reduced expression for d(t) ∈ S n . (ii) Let s ∈ Std(λ) be such that s t. If w ∈ S n satisfies wt = s then ψ w v t = v s .
Proof. By (2.9), in order to prove (i) it suffices to show that w := d(t) is fully commutative, i.e. that every reduced expression for w can be obtained from any other by using only relations of the form s i s j = s j s i for i, j ∈ [1, n − 1] with |i − j| > 1. By [1,Theorem 2.1], this is equivalent to showing that for no triple 1 i < j < k n it is the case that wi > wj > wk. This last statement is clear since two of i, j, k must lie in the same column of t. Part (ii) follows from (i) because a reduced expression for d(s) can be obtained by concatenating reduced expressions for d(t) and w, by Lemma 5.3.
We note that the statements of Lemmas 5.3 and 5.4 are generally false for partitions with arbitrarily many columns. An analogue of Lemma 5.4(i) for 2-row partitions is [19,Lemma 3.17].
In the rest of the section, we assume that O = Z and write R Λ 0 n := R Λ 0 n,Z . The results proved over Z below may be seen to hold over an arbitrary commutative ring by extending scalars.
Proof. By Theorem 4.8 and Lemma 4.7(ii), the R Λ 0 n,Q -module S λ Q has exactly two graded composition factors, namely D λ Q and D µ Q 1 , so there is a non-zero homomorphism of degree 1 from S µ Q to S λ Q . It is easy to see that t λ is the only standard λ-tableau with residue sequence i µ , so a scalar multiple of this homomorphism sends v µ to v t λ . In view of Corollary 2.2, the lemma follows.
Consider a partition λ = (2 x , 1 y ) ∈ Par 2 (n) such that y ≡ −j − 1 (mod e) and x j for some 1 j < e. We define T λ e to be the tableau obtained by putting t (2 x−j ) on top of the tableau t (2 j ,1 y ) , with all entries in the latter tableau increased by 2(x − j). In other words, for all (a, b) ∈ λ , we set if a > x − j and b = 2.
Proof. This follows from Lemma 5.5 and Theorem 5.1 applied x − j times.
Remark 5.7. Up to scalar multiples, the only non-zero homomorphisms between Specht modules S λ Q and S µ Q for distinct λ, µ ∈ Par 2 (n) are obtained from those given by Theorem 5.6 by extending scalars. This follows from a consideration of composition factors of these Specht modules given by Theorem 4.8.
The aim of the rest of this section is to describe the kernel and image of the homomorphism ϕ λ,µ from Theorem 5.6 in terms of e-regular tableaux.
That is, a standard λ-tableau t lies in Q r if and only if the path π t hits the wall H m at step 2r + me − 1 and does not hit any walls at later steps. Then If e = 2, then j = 1 and Q r = ∅ for all r < x − j. If e > 2, then each set Q r on the right-hand side of (5.1) is non-empty.
Lemma 5.8. Suppose that 0 r x − j and that t, s ∈ Q r satisfy t↓ 2r+me−1 = s↓ 2r+me−1 . If w ∈ S n is such that s = wt, then v s = ψ w v t .
Proof. First, note that we can turn t into s by a series of elementary transpositions of the form s a with 2r + me − 1 < a < n. Hence, it suffices to prove the lemma when w = s a = (a, a + 1) is an elementary transposition with a > 2r + me − 1.
If a + 1 lies in the first column of t and a lies in the second, then t s, so v s = ψ a v t by Lemma 5.4(ii), as required. So we may assume that a lies in the second column of s, whence it follows that v t = ψ a v s . Let k and l be determined from s(a) = (k, 2) and s(a + 1) = (l, 1), and set i := i s . Then (m − 1)e − 1 < π s (a) = l − k − 1 < π s (a + 1) = l − k < π t (a) = l − k + 1 < me − 1.
Lemma 5.10. Assume that e > 2. Suppose that 0 r x − j and t ∈ Q r . If t = wT r , where w ∈ S n , then v t = ψ w v Tr .
Proof. Let i = i Tr and i = s 2r+me i, so that (1 − 1 i )ψ 2r+me v Tr = 0. It suffices to show that {t ∈ Std(λ) | i t = i } = ∅, for then 1 i S λ Z = 0. Suppose for contradiction that t ∈ Std(λ) has residue sequence i . First, we claim that t(a) = T r (a) whenever 2r + me + 2 a n. Assuming the claim to be false, we choose a to be maximal such that the equality fails. By maximality of a, Shape(t↓ a ) = Shape(T r ↓ a ) =: (2 c , 1 d ) for some c, d ∈ Z 0 . Since a lies in different columns in t and T r and i t a = i Tr a , the residues of the bottom nodes of the two columns of (2 c , 1 d ) must be equal. However, since (m − 1)e − 1 < π Tr (a) < me − 1, we have d ≡ −1 (mod e), from which it follows that these two residues are not equal. This contradiction proves the claim.
Therefore, res t(2r + me + 1) = 1 − r + eZ. However, t(2r + me + 1) must be the bottom entry of either the first or the second column of γ, and these two entries have residues 2 − r + eZ and −r + eZ respectively, a contradiction.
Proposition 5.13. If t ∈ Std(λ) \ DStd e (λ) then v t lies in the R Λ 0 n -submodule of S λ Z generated by v T λ e .
Assume that e > 2. Let U be the submodule in the statement of the proposition. By Lemma 5.10, it is enough to show that v Tr ∈ U for all 0 r x − j. We use backward induction on r and note that v T x−j = v T λ e ∈ U , so we may assume that r < x − j and v T r+1 ∈ U . By the inductive hypothesis and Lemma 5.10 applied again, S r+1 ∈ U . Hence, v Tr ∈ U by Lemma 5.12.
The hypotheses of the following theorem are the same as in Theorem 5.6.
Proof. We use backward induction on y. Note that the kernel and image of ϕ λ,µ are free Z-modules by Proposition 2.1. By Proposition 5.13, where the penultimate equality holds because, by Lemma 3.6, the map reg e restricts to a bijection Std(λ) \ DStd e (λ) ∼ −→ DStd e (µ). We claim that ker(ϕ λ,µ ) ⊇ v t | t ∈ Std(µ) \ DStd e (µ) Z . (5.4) Let m ∈ Z >0 be such that y = me − j − 1. If x < e, then for all t ∈ Std(µ) and 0 a n we have π t (a) / ∈ H m+1 , from which it follows that t ∈ DStd e (µ) and (5.4) holds trivially, as its right-hand side is 0. So, suppose that x e and let ν := (2 x−e , 1 y+2e ), noting that y + 2e is the image of y + 2j under reflection with respect to H m+1 . By the inductive hypothesis, the right-hand side of (5.4) is exactly the image of ϕ µ,ν . Now ϕ λ,µ ϕ µ,ν = 0 because, by Theorem 4.8, S λ Q and S ν Q have no composition factors in common. Thus, the claim follows. Using (5.3) and (5.4), we obtain the first equality in the theorem. Also, (5.3) is an exact equality, so the two sides of (5.2) have the same Z-rank. This completes the proof since the right-hand side of (5.2) is a pure Z-submodule of S λ Z . Remark 5.15. Let λ ∈ Par 2 (n). It is not difficult to show that if t ∈ DStd e (λ) and s ∈ Std(λ) \ DStd e (λ) then i t = i s (cf. the claim in the proof of Lemma 5.11). This leads to a more direct proof of the first equality in Theorem 5.14.
Proof of Theorem 1.1. Let λ = (2 x , 1 y ), where y = me − 1 − j for some m ∈ Z >0 and 0 j < e. If j = 0 or x < j, then DStd e (λ) = Std(λ) and parts (i) and (iii) of the theorem hold. Also, in this case S λ K = D λ K by Theorem 4.8 and Lemma 4.7, so (ii) holds as well. On the other hand, if j = 0 and x j, then (i) and (iii) follow from Theorems 5.6 and 5.14, and (ii) again follows from Theorem 4.8.
Remark 5.16. Theorem 1.1(ii) is true with K replaced by any field of characteristic 0.
Recall the reflections s m from §3.3: we have s m · (me − 1 + j) = me − 1 − j for all m ∈ Z >0 and j ∈ Z.

Partitions with more than two columns
In this section, we outline a natural approach to extending the definition of DStd e (λ) to the case when a partition λ has more than two columns and give an example showing that this approach does not always work. For simplicity, we consider algebras over Q only, though all statements below are true with Q replaced by any field of characteristic 0.
Fix n ∈ Z 0 and λ ∈ Par(n). Given t ∈ Std(λ) and u ∈ S λ Q , we say that u is a t-element if u = v t + s∈Std(λ) s t a s v s for some coefficients a s ∈ Q. It is important to note that, whereas the elements v t depend on certain choices of reduced expressions, the set of all t-elements on S λ Q does not depend on such choices. This is the case because if one changes the reduced expression for d(t), causing v t to be replaced by v t 1 , then v t 1 = v t + s∈Std(λ), s t a s v s for some coefficients a s ∈ Q (by [5,Proposition 4.7]).
Recall from §2.4 the bilinear form ·, · on S λ Q and its radical rad S λ Q . Note that, by the properties of the form stated after (2.17), for any t, s ∈ Std(λ), we have v t , v s = 0 unless deg(t) + deg(s) = 0 and i t = i s . These facts are used repeatedly in the sequel. Define IStd e (λ) := {t ∈ Std(λ) | rad S λ Q contains at least one t-element}.
For each t ∈ IStd e (λ), choose a t-element w t ∈ rad S λ Q . The set {w t | t ∈ IStd e (λ)} is always linearly independent over Q. In the sequel, we call the partition λ e-agreeable if {w t | t ∈ IStd e (λ)} is a basis of rad S λ Q or, equivalently, if | IStd e (λ)| = dim(rad S λ Q ). Whether or not {w t | t ∈ IStd e (λ)} spans rad S λ Q does not depend on the choice of the elements w t . Note that for each t ∈ Std(λ) one can choose w t in such a way that w t is homogeneous of degree deg e (t) and 1 i t w t = w t .
Whenever λ is e-agreeable, it is reasonable to define DStd e (λ) as the complement Std(λ) \ IStd e (λ). Then ch q DStd e (λ) = ch q D λ Q thanks to the last observation in the previous paragraph. Theorem 1.1 shows that, if λ ∈ Par 2 (n), then λ is e-agreeable and the definition of DStd e (λ) just given agrees with the combinatorial one in §3.3. {B st | s, t ∈ Std(µ)} of R Λ 0 n,Q , which does not depend on any choices of reduced expressions. This new cellular structure yields a basis {B t | t ∈ Std(λ)} of S λ Q . It follows from [14, Proposition 6.7] that B t is a t-element for each t ∈ Std(λ).
A conjecture of Mathas [23, Conjecture 4.4.1] implies, in particular, that for every λ ∈ Par(n) there is a subset T λ of Std(λ) such that {B t | t ∈ T λ } is a basis of rad S λ Q . Since each B t is a t-element, this in turn implies (via elementary linear algebra) that every partition λ is eagreeable. However, Example 6.2 below shows that (for e = 3) not all partitions are e-agreeable. Hence, there is a counterexample to [23, Conjecture 4.4.1].
Hence, v t 1 , v s = 1 = v t 2 , v s . Therefore, the degree 2 component of 1 i (rad S λ Q ) is 1dimensional and is spanned by v t 1 − v t 2 . However, since neither of t 1 and t 2 dominates the other, neither of these two tableaux belongs to IStd e (λ). Hence, λ is not e-agreeable.