Sign sequences and decomposition numbers

We obtain a closed formula for the $v$-decomposition numbers $d_{\lambda\mu}(v)$ arising from the canonical basis of the Fock space representation of $U_v(\hat{\mathfrak{sl}}_e)$, where the partition $\lambda$ is obtained from $\mu$ by moving some nodes in its Young diagram, all of which having the same $e$-residue. We also show that when these $v$-decomposition numbers are evaluated at $v=1$, we obtain the corresponding decomposition numbers for the Schur algebras and symmetric groups.


Introduction
Despite its rich structure, symmetric group algebras in positive characteristic are still not well understood. Many problems -some of them fundamental -remain open. Among them, one of the most famous is the complete determination of its decomposition numbers. This has been shown to be equivalent to many other open problems, such as the complete determination of the dimensions of the irreducible modules of symmetric groups, or the complete determination of the decomposition numbers of the (classical) Schur algebras.
Algebras related to the symmetric group algebras includes the abovementioned Schur algebras, the q-Schur algebras and the Iwahori-Hecke algebras. Also related is the Fock space representation F of U v ( sl e ). The v-decomposition numbers d λµ (v) arising from the canonical basis of F is shown to give the corresponding decomposition numbers of the q-Schur algebras at complex e-th root of unity when evaluated at v = 1 (see [9]). As the decomposition matrix of Schur algebras in characteristic p can be obtained from that of the q-Schur algebras at complex p-th root of unity by postmultiplying the latter by an adjustment matrix (see [4]), d λµ (v)| v=1 may be regarded as a first approximation to the decomposition number d λµ of the Schur algebra.
While there exist several algorithms to compute the v-decomposition numbers d λµ (v) -which are parabolic Kazhdan-Lusztig polynomials and are of independent interest -these are all inherently recursive in nature, and in practice can only be applied to 'small' cases. As such, it is desirable to have closed formulae. While this may seem too ambitious in general, one can hope for such closed formulae when λ and µ are related in some way.
In the same way, while we are currently far from a complete solution to the determination of decomposition numbers d λµ of symmetric groups and Schur algebras, one can look for partial solutions when λ and µ are related in some way. In this regard, Kleshchev [6] described d λµ when λ is obtained from µ by moving one node in its Young diagram. He introduced the sign sequence induced by the pair (λ, µ) and showed that the decomposition number d λµ equals the number of latticed subsets for this sign sequence. His approach was to study the Schur algebra and the symmetric group algebra via the special linear group as an algebraic group, and the Schur functor. An independent and more elementary combinatorial approach was used in [2] to show that the corresponding v-decomposition numbers is a sum of monic monomials indexed by the latticed subsets.
It is natural to ask if one can extend the results of [6] and [2], i.e. to determine d λµ and/or d λµ (v) when λ is obtained from µ by moving k nodes, where k > 1. We provide an affirmative answer in this paper for the case where the k nodes moved all have the same residue. In this case, our main result shows that the v-decomposition number d λµ (v) is a sum of monic monomials indexed by the well-nested latticed paths for the sign sequence induced by (λ, µ), and that the corresponding decomposition number d λµ can be obtained by evaluating the v-decomposition number d λµ (v) at v = 1, and hence equals the number of well-nested latticed paths. We believe the reader will appreciate the simple definition of (well-nested) latticed paths, as compared to the rather technical one of latticed subsets.
Our approach is combinatorial, similar to that of [2], though we keep the paper self-contained and independent of [2]; in fact the results in [2] are reproved here. We obtain the above-mentioned v-decomposition numbers as well as the analogue of some relevant branching coefficients for the Fock space in this way. Assuming Kleshchev's results [6] on the branching coefficients of the form [Res S n−1 L λ : L µ ] where µ is obtained from the partition λ of n by removing a normal node, we then show that these v-decomposition numbers give the corresponding decomposition numbers for the Schur algebras and the symmetric groups when evaluated at v = 1. We also obtain the relevant branching coefficients for the Schur algebras and the symmetric groups.
The paper is organised as follows: we give a summary of the background theory in the next section. In Section 3, we set up the machineries with which we can state our main results. In Section 4, we state and prove the main results, except for Theorem 4.4 where we postpone its proof to the next section. In Section 5, we show that Theorem 4.4 is equivalent to Theorem 5.1, a combinatorial statement involving sign sequences and wellnested latticed paths. We then prove Theorem 5.1, thereby completing the proof of Theorem 4.4.

Preliminaries
In this section, we give a brief account of the background theory that we shall require. From now on, we fix an integer e ≥ 2. Also, Z e = Z/eZ, and N 0 is the set of non-negative integers.
2.1. Partitions. Let n ∈ N 0 . A partition λ of n is a finite weakly decreasing sequence of positive integers summing to n. Denote the set of partitions of n by P n , and let P = n∈N 0 P n . If λ = (λ 1 , . . . , λ s ), we write l(λ) = s.
The elements of [λ] are called nodes of λ. If r ∈ Z e and (i, j) is a node of λ such that j − i ≡ r (mod e), then we say that (i, j) has e-residue r, and that for some partition µ, in which case, we also call (i, j) an indent node of µ. A node (i 1 , j 1 ) is on the right of another node (i 2 , j 2 ) if j 1 > j 2 .
Let t ∈ Z with t ≥ l(λ). The set is the β-set of λ of size t.

2.2.
The Jantzen order and a more refined pre-order. Let λ, τ ∈ P.
We write λ ∼ r τ if and only if s λ,r,t = s τ,r,t for some t ≥ l(λ), l(τ ) (or equivalently, for all t ≥ l(λ), l(τ )). Then ∼ r is an equivalence relation on P; we write t λ r for the equivalence class containing λ. We define a total order on P/∼ r as follows: t λ r > t τ r if and only if there exist t and j such that s λ,r,t (i) = s τ,r,t (i) for all i > j, and s λ,r,t (j) > s τ,r,t (j). Lemma 2.1. Let r ∈ Z e , and let λ, τ ∈ P. If λ ≥ J τ , then t λ r ≥ t τ r . Proof. It is straightforward to verify that λ → µ implies t λ r > t µ r .
2.3. q-Schur algebras. Let F be an algebraically closed field of characteristic l, where l is either zero, or e (if e is prime), or is coprime to e. Let q ∈ F * be such that e is the least integer such that 1 + q + · · · + q e−1 = 0 (i.e. q is a primitive e-th root of unity if l = 0 or l is coprime to e, and q = 1 if l = e). The q-Schur algebra S n = S F,q (n, n) over F has a distinguished class {∆ µ | µ ∈ P n } of modules called Weyl modules. Each ∆ µ has a simple head L µ , and the set {L µ | µ ∈ P n } is a complete set of pairwise non-isomorphic simple modules of S n . (Note that ∆ µ and L µ are denoted as W µ ′ and F µ ′ respectively in [3].) The projective cover P µ of L µ (or of ∆ µ ) has a filtration in which each factor is isomorphic to a Weyl module; the multiplicity of ∆ λ in such a filtration is well-defined, and is equal to the multiplicity of L µ as a composition factor of ∆ λ . We denote this multiplicity as d l λµ , which is a decomposition number of S n . The Jantzen sum formula (see, for example, [5]) provides the following consequence: The decomposition numbers in characteristic l and those in characteristic 0 are related by an adjustment matrix A l : let D l = (d l λµ ) λ,µ∈Pn , then D l = D 0 A l . Furthermore, the matrices A l , D l and D 0 are all lower unitriangular with nonnegative entries when the partitions indexing its rows and columns are ordered by a total order extending the dominance order on P n . As a consequence, we have The Schur functor maps the Weyl module ∆ µ to the Specht module S µ of the Iwahori-Hecke algebra H n = H F,q (n). It also maps the simple module L µ of S n to the simple module D µ of H n if µ is e-regular, and to zero otherwise. As such, the decomposition matrix of H n is a submatrix of S n .
We note that if q = 1 (or equivalently, l = e), then S n is the classical Schur algebra and H n is the symmetric group algebra FS n .

Restriction and induction.
The q-Schur algebra S n is isomorphic to a subalgebra of S n+1 so that the restriction functor Res Sn and the induction functor Ind S n+1 make sense. In fact For each r ∈ Z e , r-Res Sn and r-Ind S n+1 are exact, preserve direct sums and are adjoint of each other. Their effects on Weyl modules can be easily described in the Grothendieck group: Theorem 2.4. Let r ∈ Z e , λ ∈ P n and τ ∈ P n+1 . Then where the first sum runs over all σ ∈ P n that can be obtained from τ by removing a removable r-node while the second sum runs over all µ ∈ P n+1 that can be obtained from λ by adding an indent r-node.
, d, d −1 subject to some relations (see, for example, [7, §4]) which we shall not require. An important U v ( sl e )-module is the Fock space representation F, which as a C(v)-vector space, has P as a basis. For our purposes, an explicit description of the action of f r on F will suffice.
Let λ be a partition and suppose that the partition µ is obtained by adding an indent r-node n to λ. Let N > (λ, µ) be the number of indent rnodes on the right of n minus the number of removable r-nodes on the right of n. We have where the sum runs over all partitions µ that can be obtained from λ by adding an indent r-node.
In [8], Leclerc and Thibon introduced an involution x → x on F, having the following properties (among others): There is a distinguished basis {G(λ) | λ ∈ P} of F called the canonical basis, which can be characterised ([8, Theorem 4.1]) as follows: Let −, − be the inner product on F with respect to which its basis P is orthonormal.
We collate together some well-known properties of d λµ (v).
Proof. Part (i) follows from [7, 7.2], parts (ii, iii) are proved by Varagnolo and Vasserot [9], part (iv) follows from parts (ii, iii) and Lemma 2.2, and part (v) is a special case of [1, Theorem 1(1)]. Theorem 2.5(iii) in particular establishes the link between the v-decomposition numbers of the Fock space and the decomposition numbers of q-Schur algebras in characteristic zero. The canonical basis vector G(µ) of F thus corresponds to the projective cover P µ of q-Schur algebras, while the standard basis element λ of F corresponds to the Weyl module ∆ λ . Under this correspondence, the action of f r on F corresponds to that of r-Ind on the q-Schur algebras.

Combinatorial setup
In this section, we introduce the notations and set up the combinatorial machineries necessary for this paper.
3.1. Notations. Given a subset A of a totally ordered set (U , ≥), and c, d ∈ U with c < d, write We also define A ≤d and A ≥c analogously.
3.2. Pairings. Given two disjoint finite subsets A and B of a totally ordered set (U , ≥), we define its associated path, denoted as Γ(A→B), as follows: For each i, represent s i by the following line segment: Connect these k line segments s 1 , s 2 , . . . , s k in order, to obtain the path Γ(A→B).
We define a pairing of A to B using Γ(A→B). An element x ∈ A is paired with π A→B (x) ∈ B that is represented by the next on the right of and at the 'same level' as the representing x, if such a exists. If such a does not exist, then x is said to be unpaired.
Denote the subsets consisting of paired and unpaired elements of A by P A→B (A) and U A→B (A), and the subsets consisting of paired and unpaired elements of B by P A→B (B) and U A→B (B). Clearly, π A→B : P A→B (A) → P A→B (B) is a bijection.
We We If P A→B (B) = B and P A→B (A) = A, we write A ↔ B. In this case, π A→B is a bijection from A to B.
When A and B are finite subsets of the totally ordered set (U , ≥) but are not necessarily disjoint, let A * = A \ B and B * = B \ A. We extend the pairing π A * →B * to π A→B by making every element of A ∩ B paired with itself. The subsets P A→B (A), U A→B (A), P A→B (B) and U A→B (B) are defined according to the pairing π A→B (thus, for example, P A→B (A) = P A * →B * (A * ) ∪ (A ∩ B)). Note then that A ։ B and A ↔ B if and only if Furthermore, let Thus each element of V (T ) indexes the at a 'valley' in Γ(T ) where the subpath to its right is a Dyck path.
The latticed paths for T are defined as follows: Γ(T ) is the generic latticed path for T , and all other latticed paths for T are connected paths obtained by replacing one or more 'ridges' in the generic latticed path by horizontal line segment(s). Denote the set of latticed paths for T by L(T ), and for each γ ∈ L(T ), define its norm γ as one plus the total number of and in it.

Example. Suppose that Γ(T ) is
The following are all the non-generic latticed paths for T : The norm of the generic latticed path is 10, while the norms of the nongeneric ones are 8, 8, 6, 4 respectively.
For a, b ∈ T ± with a < b, we define the following sign subsequences of T : For convenience, we define the empty path to be the only latticed path for T a a , with zero norm. Let A, B ⊆ T ± such that A ↔ B, and let π = π A→B . A well-nested latticed path for (T, A, B) is a collection (γ a ) a∈A such that for each a ∈ A, γ a ∈ L(T π(a) a ), and no part of γ a falls below the subpath of γ a ′ between a and π(a) whenever a ′ < a < π(a) < π(a ′ ). We denote the set of well-nested latticed paths for (T, A, B) by Ω(T B A ), and for each ω = (γ a ) a∈A ∈ Ω(T B A ), we define its norm ω as a∈A γ a .
Remark. Our definition of sign sequence generalises the original one used by Kleshchev in [6] (and subsequently used in [2]). His definition [6, Definition 1.1] corresponds naturally to our sign sequences T with T ± = {1, 2, . . . , |T ± |}, naturally ordered. For such a sign sequence T , and γ ∈ L(T ), let Then X γ is a latticed subset for T in the sense of [6, Definition 1.8]. Furthermore, γ = 1 + 2|X γ | + |T |. Conversely, given a latticed subset X for T , one can find a unique γ ∈ L(T ) such that X γ = X. There is thus a one-to-one correspondence between latticed paths and latticed subsets. We leave the proof of these facts as a combinatorial exercise for the reader.

Main results
We state and prove the main results in this section, with the exception of Theorem 4.4, whose proof is postponed to the next section.
Let λ ∈ P, and let r ∈ Z e . Denote the sets of removable r-nodes and indent r-nodes of λ by R r (λ) and I r (λ) respectively. Let T r (λ) be the sign sequence (R r (λ), I r (λ)), where R r (λ) ∪ I r (λ) is totally ordered as follows: a > b if and only if a is on the right of b.
For X ⊆ I r (λ) and Y ⊆ R r (λ), we write λ↑ X ↓ Y for the partition obtained from λ by adding all the indent r-nodes x with x ∈ X and removing all the removable r-nodes y with y ∈ Y . We extend this notation to all subsets X, Y ⊆ R r (λ) ∪ I r (λ) such that X \ Y ⊆ I r (λ) and Y \ X ⊆ R r (λ), i.e. for these subsets, λ↑ X ↓ Y = λ↑ X\Y ↓ Y \X .
We note that t λ r = t µ r (see Section 2.2) if and only if µ = λ↑ X ↓ Y for some X ⊆ I r (λ) and Y ⊆ R r (λ). Lemma 4.1. Let λ ∈ P n and r ∈ Z e . Let A ⊆ I r (λ) and B ⊆ R r (λ) with Proof. If d λ↑ A ↓ B ,λ (v) = 0 or d l λ↑ A ↓ B ,λ = 0, then λ ≥ J λ↑ A ↓ B by Theorem 2.5(iv) and Lemma 2.2. In particular, λ dominates λ↑ A ↓ B , so that A ։ B.
Let λ ∈ P n and r ∈ Z e . The induced module r-Ind S n+1 P λ is projective, so that where a l τ λ ∈ N 0 satisfies We have an analogue of Proposition 4.2 for q-Schur algebras.
Proposition 4.3. Let λ ∈ P n and r ∈ Z e . Let r-Ind P λ = τ ∈P n+1 a l τ λ P τ . Suppose that in the Grothendieck group (i) If σ ∈ P n+1 such that c l σλ = 0, then t σ r ≤ t λ r . (ii) If σ ∈ P n+1 such that a l σλ = 0, then t σ r ≤ t λ r . (iii) If A ⊆ I r (λ) and B ⊆ R r (λ) with |A| = |B| + 1, then The proof of Proposition 4.3 is entirely analogous to that of Proposition 4.2.
(ii) Suppose that f r (G(λ)) = µ a µλ (v)G(µ). Then a λ↑ A ↓ B ,λ (v) = 0 whenever |A| = |B| + 1 and A ։ B, unless A = {a} ⊆ V (T r (λ)) (and B = ∅), in which case, where here, and hereafter, for k ∈ Z + , we write Theorem 4.4 will be proved in the next section. For the remainder of this section, we shall assume Theorem 4.4 and obtain its analogue for the classical Schur algebras.
Recall that the following branching coefficient is obtained by Kleshchev; note that a node n ∈ R r (λ) is normal if and only if n ∈ U + (T r (λ)): Theorem 4.5 ([6, Theorem 9.3]). Let λ ∈ P n and r ∈ Z e . Let b ∈ U + (T r (λ)). Then Theorem 4.6. Suppose that e is a prime integer. Let λ ∈ P n and r ∈ Z e . Let A ⊆ I r (λ) and B ⊆ R r (λ).
for all X 2 and Y 2 such that (X 2 , Y 2 ) (X 1 , Y 1 ). Now let λ ∈ P n and assume that part (i) holds for all µ ∈ P n−1 . Let A ⊆ I r (λ) and B ⊆ R r (λ).

Proof of Theorem 4.4
We provide the proof of Theorem 4.4 in this section. Let λ ∈ P n and r ∈ Z e . Write T = (T + , T − ) for T r (λ) = (R r (λ), I r (λ)), and let A ⊆ T − and B ⊆ T + such that |A| = |B| + 1 and A ։ B. Consider f r (G(λ)), λ↑ A ↓ B , which can be computed in the following two ways: by (1) and Lemma 4.1. If Theorem 4.4(i) holds for λ, then we have where the sum runs over all c ∈ T + ∪ A \ B such that A ։ (B ∪ {c}) and over all ω ∈ Ω(T
We assume further that Theorem 4.4(ii) holds for λ and all C ⊆ I r (λ), D ⊆ R r (λ) such that (C, D) ≺ (A, B). By Theorem 5.1, the right-hand sides of (5) and (6) are equal. Both are equal to f r (G(λ)), λ↑ A ↓ B assuming the conjectural formulae in Theorem 4.4 hold. By our induction hypothesis, these formulae do hold for all terms involved except for a λ↑ A ↓ B ,λ (v) and d λ↑ A ↓ B ,λ↑ {x} (v), where x indexes the indent r-node on the first row of λ (if it exists; the formula holds for other d λ↑ A ↓ B ,λ↑ {d} (v) by Theorem 2.5(v)). We are thus reduced to the following equations: Let µ ∈ P n+1 , and let X ⊆ I r (µ), Y ⊆ R r (µ). If µ does not have an indent r-node on its first row, or if µ has an indent r-node on its first row, indexed by x say, with x / ∈ Y , then d µ↑ X ↓ Y ,µ (v) equals the conjectural formula of Theorem 4.4(i) by Theorem 2.5(v). If µ has an indent r-node on its first row, indexed by x, and x ∈ Y , then let λ = µ↓ {x} . We have seen from above that Proof of Theorem 5.1. We prove by induction, and consider the following three cases separately.
• B = ∅: This is the base case for the induction, and is essentially proved in [2]. More specifically, we define φ : The reader may check that both φ and ψ are well-defined bijections that reduce the norm of each element in their respective domains by 1 − |T b 0 a 0 |. By induction hypothesis (on |B|), there is a norm-preserving bijection χ : is a norm-preserving bijection. • |B| > 0, and (T + ) b a = ∅ for all a ∈ A, b ∈ B with a < b: This is the most difficult case. Let a 0 ∈ A, b 0 ∈ B with a 0 < b 0 , such that |(T + ) b 0 a 0 | ≤ |(T + ) b a | for all a ∈ A, b ∈ B with a < b. Then (T ± ) b 0 a 0 ∩ B = ∅ and, by replacing a 0 with min((T ± ) b 0 a 0 ∩ A) if necessary, we may assume that ( , and T ′ = (T + \ {b 1 }, T − \ {a 2 }). Let φ 1 : LB A (T ) → L B A (T ) be defined as follows. Let (c,ω) ∈ LB A (T ). For c = b 0 , let a 1 = π −1 A→(B∪{c}) (b 1 ). Then π A→(B∪{c}) (a 1 ) = b 0 . Let ω ∈ Ω(T B∪{c} A ) be obtained fromω = (γ a ) a∈A by replacing γ a 1 with the latticed path for T b 0 a 1 obtained by extendingγ a 1 by the generic latticed path between b 1 (inclusive) and b 0 . Define Let φ 2 : L B A (T ′ ) → L B A (T ) be defined as follows. Let (c, ω ′ ) ∈ L B A (T ′ ), with ω ′ = (γ ′ a ) a∈A . Write π for π A→(B∪{c}) . For a ∈ A, define γ a ∈ L(T π(a) a ) to be the latticed path obtained by inserting horizontal line segments into γ ′ a at positions b 1 and a 2 if a < b 1 < π(a), and as γ ′ a otherwise. Let ω = (γ a ) a∈A . Then ω ∈ Ω(T
Let ̟ 2 ∈ Ω((T ↑ d ) B∪{d} A ) be obtained from̟ by replacingδ a 0 with the latticed path for (T ↑ d ) d a 0 obtained by extendingδ a 0 by the generic latticed path between b 1 (inclusive) and d, and replacing δ a ′ with the latticed path for (T ↑ d ) b 0 a ′ obtained by extendingδ a ′ by the generic latticed path between d (inclusive) and b 0 (which is just at d). Define Let ψ 2 : R B A (T ′ ) → R B A (T ) be defined as follows. Let (d, d ′ , ̟ ′ ) ∈ R B A (T ′ ), with ̟ ′ = (δ ′ a ) a∈A . Write π for π A→(B∪{d}) . For a ∈ A, define δ a ∈ L((T ↑ d ) π(a) a ) to be the latticed path obtained by inserting horizontal line segments into δ ′ a at positions b 1 and a 2 if a < b 1 < π(a), and as δ ′ a otherwise. Let ̟ = (δ a ) a∈A . Then ̟ ∈ Ω((T ↑ d )