On homomorphisms into Weyl modules corresponding to partitions with two parts

Abstract Let K be an infinite field of characteristic 
$p>0$
 and let 
$\lambda, \mu$
 be partitions, where 
$\mu$
 has two parts. We find sufficient arithmetic conditions on 
$p, \lambda, \mu$
 for the existence of a nonzero homomorphism 
$\Delta(\lambda) \to \Delta (\mu)$
 of Weyl modules for the general linear group 
$GL_n(K)$
 . Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.


Introduction
In the representation theory of the general linear group GL n (K), where K is an infinite field of characteristic p > 0, the Weyl modules ∆(λ) are of central importance.These are parametrized by partitions λ with at most n parts.Over a field of characteristic zero, the modules ∆(λ) are irreducible.However over fields of positive characteristics this is no longer true and determining their structure is a major problem.In particular, very little is known about homomorphisms between them.
For GL 3 (K) all homomorphisms between Weyl modules have been classified when p > 2 by Cox and Parker [5].Some of the few general results are the non vanishing theorems of Carter and Payne [4] and Koppinen [11], and the row or column removal theorems of Fayers and Lyle [14] and Kulkarni [12].
In [17] we examined homomorphisms into hook Weyl modules and obtained a classification result.This has been obtained also by Loubert [13].In the present paper we consider homomorphisms ∆(λ) → ∆(µ), where µ has two parts.The main result, Theorem 3.1, provides sufficient arithmetic conditions on λ, µ and p so that Hom S (∆(λ), ∆(µ)) = 0, where S is the Schur algebra for GL n (K) of appropriate degree.An explicit map is provided that corresponds to the sum of all standard tableaux of shape µ and weight λ.The main tool of the proof is the description of Weyl modules by generators and relations of Akin, Buchsbaum and Weyman [2].
The first examples of pairs of Weyl modules with homomorphism spaces of dimension greater than 1 were obtained by Dodge [6].Shortly after, more were found by Lyle [14].In Corollary 6.2, we find sufficient conditions on λ, µ and p so that dim Hom S (∆(λ), ∆(µ)) > 1 and thus we have new examples of homomorphism spaces between Weyl modules of dimension greater than 1.
By a classical theorem of Carter and Lusztig [3], the results in Theorem 3.1 and Corollary 6.2 have analogues for Specht modules for the symmetric group when p > 2, see Remark 3.2 and Remark 6.3.
Section 2 is devoted to notation and preliminaries.In Section 3 we state the main result and in Section 4 we consider the straightening law needed later.The proof of the main result is in Section 5.In Section 6 we consider homomorphism spaces of dimension greater than 1.

Preliminaries
2.1.Notation.Throughout this paper, K will be an infinite field of characteristic p > 0. We will be working with homogeneous polynomial representations of GL n (K) of degree r, or equivalently, with modules over the Schur algebra S = S K (n, r).A standard reference here is [8].
In what follows we fix notation and recall from Akin and Buchsbaum [1], and also Akin, Buchsbaum and Weyman [2] important facts.
Let V = K n be the natural GL n (K)-module.The divided power algebra DV = i≥0 D i V of V is defined as the graded dual of the Hopf algebra S(V * ), where V * is the linear dual of V and S(V * ) is the symmetric algebra of V * , see [2], I.4.For v ∈ V and i, j nonnegative integers, we will use many times relations of the form where i+j j is the indicated binomial coefficient.By ∧(n, r) we denote the set of sequences a = (a 1 , . . ., a n ) of nonnegative integers that sum to r and by ∧ + (n, r) we denote the subset of ∧(n, r) consisting of sequences r) are referred to as partitions of r with at most n parts.The transpose partition The exterior algebra of V is denoted ΛV = i≥0 Λ i V .If a = (a 1 , . . ., a n ) ∈ ∧(n, r), we denote by Λ(a) the tensor product For λ ∈ ∧ + (n, r), we denote by ∆(λ) the corresponding Weyl module for S. In [2], Definition II.1.4,the module ∆(λ) (denoted K λ F there), was defined as the image a map particular d ′ λ : D(λ) → Λ(λ t ).For example, if λ = (r), then ∆(λ) = D r V , and if λ = (1 r ), then ∆(λ) = Λ r V .

2.2.
Relations for Weyl modules.We recall from [2], Theorem II.3.16, the following description of ∆(λ) in terms of generators and relations.

2.3.
Standard basis of ∆(µ).We will record here and in the next subsection two important facts from [2] and [1] specified to the case of partitions consisting of two parts.
Let us fix the order e 1 < e 2 < ... < e n on the set {e 1 , e 2 , ..., e n } of the canonical basis elements of the natural module V of GL n (K).We will denote each element e i by its subscript i.For a partition µ = (µ 1 , µ 2 ) ∈ ∧ + (n, r), a tableau of shape µ is a filling of the diagram of µ with entries from {1, ..., n}.Such a tableau is called standard if the entries are weakly increasing across the rows from left to right and strictly increasing in the columns from top to bottom.(The terminology used in [2] is 'co-standard').
The set of standard tableaux of shape µ will be denoted by ST(µ).The weight of a tableau T is the tuple α = (α 1 , ..., α n ), where α i is the number of appearances of the entry i in T .The subset of ST(µ) consisting of the (standard) tableaux of weight α will be denoted by ST α (µ).
To each tableau T of shape µ = (µ 1 , µ 2 ) we may associate an element and a ij is equal to the number of appearances of j in the i-th row of T .For example, the T depicted above yields x T = 1 (3) 2 (2) 4 ⊗ 2 (2) 34.
For the computations to follow, we need to make the above correspondence explicit.Let ν = (ν 1 , ..., ν n ) ∈ ∧(n, r) and T ∈ ST ν (µ).Let a i (respectively, b i ) be the number of appearances of i in the first row (respectively, second row) of T .We note that ν i = a i + b i for each i.In particular we have a 1 = ν 1 because of standardness of T .Define the map where is x sis (a s ) ⊗ x sis (b s ) ′ is the image of x s under the component of the diagonalization ∆ : DV → DV ⊗ DV of the Hopf algebra DV for s = 2, ..., n.
than a basis of the Kvector space Hom S (D(λ, ∆(µ)) is given by the elements φ T , where are such that

Main result
In order to state the main result of this paper we use the following notation.If x, y are positive integers, let If x is a positive integer, let R(x, 0) = 0.
Remarks 3.3.Here we make some comments concerning the inequalities n ≥ r, m ≥ 2 and µ 2 ≤ λ 1 ≤ µ 1 in the statement of the above theorem. ( The assumption n ≥ r is needed so that the Weyl modules ∆(λ), ∆(µ) are nonzero.As is usual with such results, it turns out that this assumption may be relaxed to n ≥ m, since m is the number of parts of the partition λ.This follows from the proof of the theorem to be given in Section 5.
For further use we note that the divisibility assumptions of Theorem 3.1 may be stated in a different way.For a positive integer y let l p (y) be the least integer i such that p i > y.From James [9], Corollary 22.5, we have the following result.

Lemma 3.4 ([9]
).Let x ≥ y be positive integers.Then p divides R(x, y) if and only if p lp(y) divides x.

Straightening
For the proof of Theorem 3.1 we will need the following identities involving binomial coefficients.Our convention is that a b = 0 if b > a or b < 0. Lemma 4.1.
(2) We proceed by induction on b 1 , the case b 1 = 0 being clear.Suppose b 1 > 0. Consider the element x ∈ D(µ 1 + b 1 , µ 2 − b 1 ), where and the map According to the analogue of Lemma II.2.9 of [2] for divided powers in place of exterior powers, we have d ′ µ (δ(x)) = 0 in ∆(µ 1 , µ 2 ).Thus where the sum ranges over all nonnegative integers j 1 , ..., j n such that j 1 +• • •+j n = b 1 , j 1 < b 1 and j s ≤ a s for all s = 2, ..., n.Let X be the right hand side of the above equality.By induction we have where the new sum ranges over all nonnegative integers k 2 , ..., k n such that in the right hand side of the above equation is equal to − j1,...,jn,k2,...,kn js+ks=is where the sum is restricted over those j 1 , ..., j n and k 2 , ..., k n that satisfy the additional conditions j s + k s = i s for all s = 2, ..., n.Hence Remembering that in the last sum we have j 1 < b 1 , Lemma 4.1(1)(b) yields j1,...,jn

Proof of the main theorem
Consider the map ψ ∈ Hom S (D(λ), ∆(µ)) given by the sum in the statement of Theorem 3.1 We will show, according to Theorem 2.1, that ψ(x) = 0 for every x ∈ Im( λ ).First we look at the relations corresponding to rows 1 and 2 of ∆(λ).
Relations from rows 1 and 2 , where t ≤ λ 2 , and let T ∈ ST λ (µ).Then T is of the form where the a i , b i satisfy the conditions of Lemma 2.3.Using the definition of φ T from 2.4, we have , then by the first part of Lemma 4.2 we obtain φ T (x) = 0. Hence we may assume that t ≤ min{λ 2 , µ 1 − λ 1 }.Using the second part of Lemma 4.2, we have in the right hand side of the last equation and let where in the third equality we used the first identity of Lemma 4.1 (2).Thus where k = k 3 + • • • + k m and the sum ranges over all nonnegative integers k 3 , ..., k m such that k ≤ b 2 and k s ≤ a s for all s = 3, ..., m.By summing with respect to T ∈ ST λ (µ) and using Lemma 2.3 we obtain ψ(x) = b2,...,bm k3,...,km where the new sum is over all nonnegative integers b 2 , ..., b m such that b in the right hand side of (5.1) and let where in the first equality we used Lemma 4.1(1)(a) and in the second equality we used the second identity of Lemma 4.1 (2).Relations from rows i and i + 1 (i > 1).This computation is similar to the previous one but simpler as there is no straightening , where i > 1 and t ≤ λ i+1 .As before let The definition of φ T yields By summing with respect to T ∈ ST(λ, µ) and using Lemma 2.3 we have where the new sum ranges over all nonnegative integers b 2 , ..., b m such that b i ≤ λ i (i = 2, ..., m) and b in the right hand side of (5.3) and let q = b i − j.The coefficient of [S] in (5.3) is equal to where in the first equality we used Lemma 4.1 (1)(a).
We have shown thus far that the map ψ = T ∈ST(λ,µ) φ T induces a homomorphism of S-modules ψ : ∆(λ) → ∆(µ) and it remains to be shown that ψ = 0. Let z = 1 (λ1) ⊗ • • • ⊗ m (λm) ∈ D(λ) and T ∈ ST λ (µ).Then from the definition of φ T we have φ T (x) = [T ] and hence The right hand side is a sum of distinct basis elements in ∆(µ) (each with coefficient 1) according to Theorem 2.2 and hence nonzero.The proof is complete.
Remark 5.1.Lyle has shown in [15], Propositions 2.19 through 2.27 and subsection 3.3, that the homomorphism spaces between Specht modules corresponding to partitions λ = (λ 1 , ..., λ n ), µ = (µ 1 , µ 2 ) of r with µ 2 ≤ λ 1 , over the complex Hecke algebra H = H C,q (S r ) of the symmetric group S r , where q is a complex root of unity, are at most 1 dimensional.Furthermore she proves exactly when they are nonzero and provides a generator which turns out to correspond to the sum of all standard tableaux in ST λ (µ).(Note that our λ, µ are reversed).In the statement of Theorem 3.1 a similar map is considered and there are some technical similarities between the proof of our main result and [15].However, we show in the next section, our modular homomorphism spaces may have dimension greater than 1.

Homomorphism spaces of dimension greater than 1
As mentioned in the Introduction, the first examples of Weyl modules ∆(λ), ∆(µ) such that dim Hom S (∆(λ), ∆(µ)) > 1 were obtained by Dodge [6].More examples were found by Lyle [14], in fact in the q-Schur algebra setting.The purpose of this section is to observe that the homomorphism spaces of Theorem 3.1 may have dimension > 1, see Corollary 6.2 and Example 6.4 below.
We recall the following special case of the classical nonvanishing result of Carter and Payne [4].Here boxes are raised between consecutive rows.See [16], 1.2 Lemma, for a proof of this particular case in our context.The main result of this section is the following.
r), we denote by D(a) or D(a 1 , . . ., a n ) the tensor product D a1 V ⊗ • • • ⊗ D an V .All tensor products in this paper are over K.