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 Dλ, labelled by e-restricted partitions λ of n, for a cyclotomic KLR algebra RnΛ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {R}_n^{\varLambda_0} $$\end{document} over a field of characteristic p ≥ 0, with mild restrictions on p. If all parts of λ are at most 2, we identify a set DStde,p(λ) of standard λ-tableaux, which is defined combinatorially and naturally labels a basis of Dλ. In particular, we prove that the q-character of Dλ can be described in terms of DStde,p(λ). We show that a certain natural approach to constructing a basis of an arbitrary Dλ 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) = 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] or §2.3. Then the Specht module S λ O has an O-basis {v t | t ∈ Std(λ)} such that 1 i v t = δ i,i t v t for any 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 we have ch q S λ K = ch q Std(λ).
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.

BASES OF SIMPLE MODULES
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. Theorem 1.1. Let λ be a 2-column partition of n. ( n,K -modules. In particular, ch q D λ K = ch q DStd e (λ). 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 Theorem 1.2. If char K = p > 0 then ch q D λ K = ch q DStd e,p (λ) for every 2column partition λ. Now suppose that char K = p 0 and let λ be a 2-column partition. James [15], [16] and Donkin [7] determined the ungraded composition multiplicities of S λ K . In particular, each D µ K appears as a composition factor of S λ K with multiplicity at most 1. Moreover, in the case when p = e (i.e., that of symmetric groups), Erdmann [9] has given a more direct description of the dimensions of simple modules labelled by e-restricted 2-column partitions: she proved that these dimensions are coefficients in certain explicitly determined generating functions. Theorem 4.10 extends the results of James and Donkin to give graded decomposition numbers. Thus, whenever D µ K appears as an (ungraded) composition factor of S λ K , there is an explicitly described integer r e,p,λ,µ ∈ {0, 1} such that D µ K r e,p,λ,µ is a graded composition factor of S λ K . Combining this fact with Theorems 1.1(ii) and 1.2, we obtain the character identity ch q S λ = µ q r e,p,λ,µ ch q DStd e,p (µ), (1.4) 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), (1.5) see 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 selfdual, 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 Zmodule 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
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.

Graded algebras and modules
By a graded module (over any ring) we mean a Z-graded one. If V is a graded module and m ∈ Z, we denote the m-th graded component 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 A-homomorphisms 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.

KLR algebras
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 sf 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: 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 Λ0 n , where Λ 0 is a (level 1) weight satisfying Λ 0 , β ∨ i = δ i,0 for all i ∈ I. The algebra R Λ0 n = R Λ0 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 Λ0 n are both Z-graded with 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 r1 . . . s rm as a product of elementary transpositions with m as small as possible. Define ψ w := ψ r1 . . . ψ rm ∈ R n , (2.12) 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 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 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,Cor. 6.26], the action of R n on S λ factors through R Λ0 n , so S λ is naturally an R Λ0 n -module. For each noting that in general v t depends on the choice of the reduced expression for d(t) made in (2.12). In particular, v t λ = v λ .

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,Thm. 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 Λ0 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].
Simple H n (ξ)-modules were classified in [6], and a corresponding classification of graded simple modules R Λ0 n,K -modules up to isomorphism and grading shift is given by [4,Thm. 4.11]. For our purposes, it is convenient to use the description of simple R Λ0 n,F -modules resulting from the graded cellular basis of R Λ0 n,F constructed by Hu and Mathas [13], which we now review. Let λ ∈ Par(n) and define Y λ := {r ∈ [1, n] | t λ (r) ∈ Z × eZ}. Set y λ := r∈Y λ y r ∈ R Λ0 n,F . There is an antiautomorphism * of R Λ0 n,F defined on the standard generators by For s, t ∈ Std(λ), set . The algebra R Λ0 n,F is a graded cellular algebra with weight poset (Par(n), ) and cellular basis {ψ st | s, t ∈ Std(λ), λ ∈ Par(n)}.
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 Λ0 n,F -module: this follows from [13, Cor. 5.10] and the proof of [19, Thm. 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 n,F -modules up to isomorphism and grading shift.
where a k is the multiplicity of D µ F k in a graded composition series of M , see [4, §2.4]. Since the algebra R Λ0 n,F is cellular, it is split, so It is well known that [S λ F : D µ F ] q = 0 unless µ λ, see, e.g., [22,Cor. 2.17]. Remark 2.6. Li [20] proved that R Λ0 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 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 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 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

Paths
Let r, s ∈ Z. Recall that [r, s] := {t ∈ Z | r t s}. We define P [r,s] to be the set of all maps π : [r, s] → V Z such that π(a + 1) − π(a) ∈ {ε 1 , . . . , ε k } for all a ∈ [r, s − 1]. Further, set Let n ∈ Z 0 . We define Given t ∈ CT k (n), for any 0 a n and 1 j k, set 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 we have 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. Proof. Let λ be the shape of t. Arguing by induction on n, we see that it is enough to show that d t(n) (λ) = deg e (π t (n − 1), π t (n)) when n > 0. For j = 1, . . . , k, let c j be the size of the jth column of λ , i.e., c j = c n,j (t). Let t be such that t(n) = (c t , t). Then i := t − c t + eZ ∈ I is the residue of t(n). It is easy to see using the definitions and (3.4) that Since π t (n) = π t (n − 1) + ε t , we have deg e,α (π t (n − 1), π t (n)) = 0 for all α ∈ Φ + such that α = ε j − ε t for any j = 1, . . . , t − 1. Moreover, using (3.1) we see that for each j = 1, . . . , t − 1, otherwise.
Remark 3.4. The correspondence between standard tableaux and paths, as above, is considered in [12,Sec. 5]. The degree function (3.6) is similar to the one defined in [2, Def. 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.
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 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: Lemma 3.6. Let µ = (2 x , 1 y ) ∈ Par 2 (n) and s ∈ Std(µ). Write y = me − 1 + j, where m ∈ Z 0 and 0 j < e. 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 if s ∈ DStd e (µ) and either y < e or j = 0; {s, t} if s ∈ DStd e (µ), y e and j > 0.
Example 3.7. Let e = 4. A path π ∈ P + 19 and the path reg 4 (π) are depicted below on the left and right, respectively. The weight space V = R is identified with a horizontal line, but-for presentation purposes-the height at which π(a) is drawn gradually increases as a = 0, . . . , n increases. (We use this convention throughout the paper.) The vertical lines indicate the walls H m = {4m − 1}, m ∈ Z 0 . In each picture, the steps from π(a) to π(a + 1) for which deg 4 (π(a), π(a + 1)) is 1 or −1 are marked by + or −, respectively; these steps are highlighted for clarity. The degree of any unmarked step is 0. respectively.

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.

(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.
(2) Otherwise, reg e,p (π) := reg e,p (reg ep z (π)), where z is the largest non-negative integer such that reg ep z (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 where, for each r = 1, . . . , h, the integer z r 0 is maximal such that the path 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
, µ∈RPar e, 2 (n) , so that D p (q) is the submatrix of the graded decomposition matrix of R Λ0 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.
Our strategy is as follows. The ungraded decomposition matrices D p for all p > 0 are known by the 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 We also define a graded version of these numbers: Note that f q e,p (l, s)| q=1 = f e,p (l, s) and f q e,p (l, 0) = 1. The ungraded decomposition numbers when p > 0 are given by the following theorem. This theorem was proved by James in the case when p = e (see [15,Thm. 24.15]) and-under certain additional conditions-when p = e (see [16,Thm. 20.6]). The result for p = e in full generality follows from a theorem of Donkin [7,Thm. 4.4(6)] together with (2.18). We follow the statement given by Mathas in [22, p. 127].
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 Our aim is to prove that [S λ F : D λ F ] q = f q e,p (y, u − x) under the hypotheses of Theorem 4.4. We do so with the aid of alternative descriptions of f q e,p (y, u − x), which are given by Lemmas 4.6 and 4.7 (for p > 0 and p = 0 respectively) and are used in the rest of the paper. First, we note the following elementary fact: Lemma 4.5. Assume that p > 0. Let a 0 , . . . , a s , l ∈ Z 0 for some s ∈ Z 0 satisfy 0 a i < p for i = 0, . . . , s and 0 l < e. If δ 0 , . . . , δ s+1 , δ 0 , . . . , δ s+1 ∈ {0, 1} and 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,   We note that the last two cases in (4.6) cannot occur simultaneously by Lemma 4.5.
( 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, Thm. 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,Thm. 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 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 2column 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.
The following lemma is an easy consequence of the definitions in §3.3. 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 is the regularisation equation of r. In particular, reg ep z (r) = r for all z > z 2 . Let m = a s p s−z1 + a s−1 p s−1−z1 + · · · + a z1 .
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. 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.

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 me − 1 < η(c) < (m + 1)e − 1 whenever r < c < s, then we call η a positive arc. • 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 subset Z = {z 1 > · · · > z h } of Z 0 , we define the tuple w(Z, η) = (w 1 , . . . , w h ) ∈ [0, n] h by setting w i to be the maximal element of [0, n] such that η(w i ) ∈ m∈Z>0 H mp z i , for each i = 1, . . . , h, with w i = 0 if no such element exists. 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.

Homomorphisms between 2-column Specht modules
Let O be an arbitrary (unital) commutative ring and n ∈ Z 0 . We recall a row removal result from [10] for homomorphisms between Specht modules, applied in the special case of the algebra R Λ0 n := R Λ0 n,O . Let λ, µ ∈ Par(n). Recall from Proposition 2.1 that S λ O has a basis {v t | t ∈ Std(λ)}. Also, recall the tableaux t µ and t µ from §2. 3 . If λ = (λ 1 , . . . , λ l ) is a partition and t ∈ Std(λ), letλ := (λ 2 , . . . , λ l ) and let t ∈ Std(λ) be the tableau obtained from t by removing the first row and decreasing all entries by λ 1 . . Let λ, µ ∈ Par(n) be such that λ 1 = µ 1 . There is an isomorphism for some coefficients a t ∈ O, then a t vt.
(i) This result is stated in [10] in terms of column Specht modules when O is a field. However, the proof works for an arbitrary commutative ring O and can be translated to the present set-up by transposing all partitions and tableaux. The second assertion of Theorem 5.1 is clear from the proof of [10,Thm. 4.1].
(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 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,Thm. 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,Lem. 3.17].
In the rest of the section, we assume that O = Z and write R Λ0 n := R Λ0 n,Z . The results proved over Z below may be seen to hold over an arbitrary commutative ring by extending scalars.
Lemma 5.5. Suppose that n ≡ j − 1 mod e with 1 j < e and n > j − 1. Let µ = (1 n ) and λ = (2 j , 1 n−2j ). Then there is a dominated R Λ0 n -homomorphism ϕ : S µ Z → S λ Z given by v µ → v t λ . Moreover, ϕ is homogeneous of degree 1. Proof. By Theorem 4.8 and Lemma 4.7(ii), the R Λ0 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 Theorem 5.6. Let λ = (2 x , 1 y ) ∈ Par 2 (n). Suppose that y ≡ −j − 1 (mod e) for some 1 j < e and that x j. Let µ = (2 x−j , 1 y+2j ). Then there is a dominated R Λ0 n -module homomorphism which is homogeneous of degree 1.
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. Since t, T r ∈ Q r , we can write w = w 1 w 2 , where w 1 fixes the set [2r +me, n] pointwise and w 2 fixes [1, 2r + me − 1] pointwise. Set t 1 := w 1 T r . By definition of T r , we have T r ↓ 2r+me−1 = t ν r , where ν r = (2 r , me − 1). Hence, T r t 1 , so by Lemma 5.4(ii) we have ψ w1 v Tr = v t1 . Lemma 5.8 yields ψ w2 v t1 = v t , whence the result follows. 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. Proof. It is easy to see that S r+1 T r . So, by Lemma 5.4(ii), we have v Sr+1 = ψ 2r+me ψ 2r+me+1 v Tr . Further, note that res T r (2r + me) = res T r (2r + me + 2) = res T r (2r + me + 1) + 1 = −r + 1 + eZ. Hence, where we have used (2.11) for the second equality and Lemma 5.11 for the third equality.
Proposition 5.13. If t ∈ Std(λ) \ DStd e (λ) then v t lies in the R Λ0 n -submodule of S λ Z generated by v T λ e . Proof. If e = 2, then T λ e dominates every element of Q x−j = Std(λ) \ DStd e (λ) (cf. the proof of Lemma 5.10), so the result follows from Lemma 5.4(ii).
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 Tx−j = v T λ e ∈ U , so we may assume that r < x − j and v Tr+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 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,Prop. 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 eagreeable 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 .