Weyl Modules Research Articles

A note on composition factors of weyl modules

Keywords: simple algebraic group ; reducibility ; Weyl modules Reference CTG-ARTICLE-1989-001doi:10.1080/00927878908823771 Record created on 2008-12-16, modified on 2017-05-12

Characteristic-free representation theory of the general linear group II. Homological considerations

In our first paper of this series Cl], we indicated how we were led to consider resolutions of Schur and Weyl modules of a particular form. In order to prove the existence of these resolutions, we were forced to enlarge the family of skew shapes to a class J containing new shapes whose corresponding Schur and Weyl modules had heretofore not been studied. With these shapes in hand, we proved in [ 1 ] the existence of some fundamental exact sequences and described how, from these exact sequences, we could use a mapping cone construction to build up the resolutions we were seeking. For this mapping cone construction, we needed maps, and to provide maps we needed projectivity of tensor products of divided powers. This led to the study of the Schur algebra and its decomposition into orthogonal idempotents. In Sections 1 and 2 we review the information about the Schur algebra that we need to carry out our program. Fortunately there is a very clear and detailed exposition of this subject in the notes of J. A. Green [S] from which we borrowed very heavily.’ In fact, the main function of the first two sections is to condense and translate into our notation and terminology the relevant sections of Green’s notes. In Sections 3 and 4 we define the family of shapes, J, that we will study,

The Euler character and cancellation theorems for Weyl modules

The Euler character and cancellation theorems for Weyl modules

A filtration of Weyl modules for large weights

AbstractA filtration is constructed for each dual Weyl module of a connected reductive group in prime characteristic p, and the quotients of the filtration are identified when the highest weight is far enough from the walls of the dominant chamber. The existence of certain composition factors is deduced.

On homomorphisms between weyl modules for hook partitions

Let K be an infinite field of finite characteristic p. We denote by Sk (n,r) the Schur Algebra [4. §4] for n and r. The aim of this paper is to determine Hom Sk(n,r)( Va V a-1) where Va is the Weyl module associated with the hook partition

Weyl modules for p-singular weights

Weyl modules for p-singular weights

The Littlewood-Richardson rule for Coschur modules

In the last 15 years, many mathematicians have turned their attention to the problem of extending to more general settings (fields of arbitrary characteristic, commutative rings) the representation theory of the general linear and symmetric groups over fields of characteristic zero. In this undertaking, the first problem is, perhaps, that of finding suitable generalizations of those modules that, in the classical theory, yield the irreducible representations first discovered and classified by Schur in his Dissertation of 1901 [24]. Over fields of characteristic zero, the more popular constructions of these modules, in terms of symmetry classes of tensors, are those due to Schur [25] and Weyl [31]; as is well known, these contructions are equivalent since they give rise to modules that, even though they are different as subspaces of the tensors space, turn out to be isomorphic with respect to the action of the group. Recently, it has been recognized that such constructions can be adequately adapted in order to make sense over arbitrary commutative rings; Akin, Buchsbaum and Weyman [ 1 ] have, therefore, introduced the so-called Schur and Coschur functors that turn out to be the natural generalizations to commutative rings of the constructions of Schur and Weyl, respectively. These functors are universally free functors and they give rise to modules-Schur and Coschur modules-which, even though they are isomorphic over fields of characteristic zero, are far from being isomorphic over the integers. Furthermore, Schur and Coschur modules are indecomposable, but they are not, in general, irreducible over commutative rings. Working from a slightly different point of view, Carter and Lusztig [7] discovered a class of modules which are irreducible over fields of positive characteristic; these modules were later elegantly described by Clausen [S] as the images of a natural operator-the Capelli operator-whose domain and codomain are the so-called Weyl modules of the second and first kind, which turn out to be the same as Coschur and Schur modules, respectively. Alongside the classification of the irreducible modules lies the second 143

On Schur algebras and related algebras, II

This paper is a continuation of [7]. The purpose of [7] was to set up, in the context of reductive algebraic groups, a theory of “generalized Schur algebras.” A special case of our construction gives the Schur algebras S(n, r) studied extensively in [lo]. We concentrate here on the Schur algebras themselves and show that our construction yields new results in their representation theory. By means of the Schur functor (see [ 10, Chap. 6]), we obtain new results in the representation theory of the symmetric groups. The main results are as follows. The Schur algebras S,(n, r), defined over a principal ideal domain R, have finite global dimension (see also [ 1 I). Over a field or complete valuation ring R, every finitely generated projective S,(n, r)-module has a Weyl module filtration. We prove the irreducibility of the socle of certain Weyl modules (see also [15]). We describe a connection between weights of projective indecomposable modules for the Schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation. We show that, over a complete valuation ring or a field, every Young module for a symmetric group has a Specht series, in particular the projective indecomposables have Specht series. Finally, we show that, for r <n and a complete valuation ring or field R, the blocks of the Schur algebra S,(n, r) are given by the rule embodied in Nakayama’s conjecture. The notations and conventions are as in the first part of this work [7].

Support Varieties of Weyl Modules

Support Varieties of Weyl Modules

Homomorphisms between neighbouring Weyl modules

Homomorphisms between neighbouring Weyl modules

A note on decomposition numbers for general linear groups and symmetric groups

In [5] James proved theorems on the decomposition numbers, for the general linear groups and symmetric groups, involving the removal of the first row or column from partitions. In [1] we gave different proofs of these theorems based on a result valid for the decomposition numbers of any reductive group. (I am grateful to J. C. Jantzen for pointing out that the Theorem in [1] may also be derived from the universal Chevalley group case, which follows from the proof of 1 ·18 Satz of [6] – the analogue of equation (1) of the proof being obtained by means of the natural isometry (with respect to contra-variant forms) between a certain sum of weight spaces of a Weyl module V(λ) of highest weight λ and the Weyl module corresponding to λ for the Chevalley group determined by the subset of the base involved.) However, we have recently noticed that this result for reductive groups, even when specialized to the case of GLn, gives a substantial generalization of James's Theorems. This generalization, which we give here, is an expression for the decomposition number [λ: μ] for a pair of partitions λ, μ whose diagrams can be simultaneously cut by a horizontal (or vertical) line so as to leave the same number of nodes above the line (or to the left of the line for a vertical cut) in both cases. Cutting between the first and second rows gives James's principal of row removal ([5], theorem 1) and cutting between the first and second column gives his principle of column removal ([5], theorem 2). Another special case of our horizontal result, involving the removal of bottom rows of a pair of partitions, is stated in [7], Satz 8.

On the structure of Weyl modules

(1984). On the structure of Weyl modules. Communications in Algebra: Vol. 12, No. 21, pp. 2665-2677.

Variations on Hulsurkar's matrix with applications to representations of algebraic Chevalley groups

Let A be an indecomposable root system for a semi-simple simply connected algebraic group defined over the field K = iF, . Let X’ be the root lattice, X the weight lattice, X+ the set of dominant weights, and W the Weyl group of A. Let Z[X] be the group ring {whose canonical basis we write in the form e(d), A E X), and let Z [X] w denote the ring of invariants under the natural action of W on Z [Xl. In [3] certain elements of Z [Xl” were defined and it was shown that they were generically equal to generalizations given by Verma [ I1 ] and Jantzen [8] of invariants defined by Hulsurkar in [4]. In this paper we will define a matrix indexed by X+ and whose entries are integers which are closely related to these elements, and from which in fact the latter can be found. We will give some properties of these elements and show how they can be computed. In particular we will show that the usual partial ordering of X+ makes this matrix upper triangular with ones on the diagonal. We will also show how our methods give some easy derivations of results of Jantzen and show how they can be used to compute generic patterns of the decomposition of Weyl modules into irreducibles.

The weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights

(1983). The weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights. Communications in Algebra: Vol. 11, No. 12, pp. 1309-1342.

On Ext1 for semisimple groups and infinitesimal subgroups

Let G be an affine, connected, semisimple, simply connected algebraic group over an algebraically closed field k of non-zero characteristic p and, for a positive integer n, let Gn be the nth infinitesimal subgroup (the kernel of the nth power of the Frobenius morphism). The purpose of this note is to point out that, for certain rational G-modules V, the natural homomorphismis the zero map, for m ≽ 0. For such modules V, taking m = 0, we see from the 5-term exact sequence of Lyndon-Hochschild-Serre that the inflation-restriction sequenceis exact and thus the dimension of H1(G, V), when finite, may be calculated by adding the dimensions of the outer terms. The practical value of this result seems to lie in the special case n = 1 and V = L(λ) ⊗ L(µ), the tensor product of simple rational G-modules. In this case H1(G/G1) VG1 is H1(G, L(λ′) ⊗ L(¯′)) for λ′, ¯′ ‘smaller’ than λ µ and the above exact sequence gives an inductive procedure for computing Ext1G between simple rational (G-modules once the (G-module structure of Ext1G1 (equi valently Ext1u1 where% is the restricted enveloping algebra of the Lie algebra of G), for simple G1-modules, is known. Using the rather complicated structure of Weyl modules Cline, in (4), calculated Ext1G between simple modules for G = SL2(k) (this process was reversed in (5)). The present note has been used by S. EIB. Yehia, (17), in finding the corresponding result for SL3(k). We believe these Ext groups to be of increasing importance in view of their connection with Lusztig's conjecture, (14), on the characters of the simple rational G-modules and the related conjecture, 7–2 of (l), of Andersen.

On the composition factors of Weyl modules.

On the composition factors of Weyl modules.

On the structure of the cohomology of line bundles on [formula omitted

One of the most intriguing problems in characteristic p representation theory for semisimple algebraic groups is the problem of finding the formal characters of the irreducible modules. An equivalent problem is to find the composition factors of Weyl modules. It is a (non-trivial) fact that the Weyl modules occur as the top cohomology groups of certain line bundles on homogeneous spaces. One could therfore more generally ask for the composition factors of all such cohomology groups. References [2] and [3] deal with aspects of this question. In the present paper we use a method primarily due to J. C. Jantzen [ 151 to extend the results in the abovementioned works. Our main results are (i) a translation principle (Theorem 2.5 below) generalizing Jantzen’s theorem in [ 15’1, (ii) a proof of the Carter-Lusztig conjecture (see [6, p. 2391, compare also [3, Theorem 4.11 and [ 71) on the existence of intertwining homomorphisms between Weyl modules (Theorem 3.3 below) together with some related results on higher Ext,-groups, and (iii) a rather general vanishing theorem for cohomology groups of line bundles on homogeneous spaces (Theorem 4.1 below). As will be apparent from a comparison with [ 151, many of Jantzen’s arguments in the proof of his translation principle can be directly carried over to our situation. However, the fact that we deal with all cohomology groups (not just the highest) allows us sometimes to shorten the proofs. In particular this is, the case for the proof of Proposition 2.3 (in which we give the effect of the translation functor on irreducible modules). In an earlier preprint [4], we showed that the vanishing theorem for dominant line bundles on homogeneous spaces can be proved (for p big) using “translation arguments.” Shortly after that preprint was written, the author 151 (and, independently, W. Haboush [ 111) found an even simpler way of proving that result. In order to describe the vanishing behaviour of the cohomology of non-ample line bundles the method is, however, still useful. This is demonstrated in the last section of this paper, where we 245 0021.8693/81,‘070245-14$02.00/O

Weyl Modules and the Cohomology of Chevalley Groups

Weyl Modules and the Cohomology of Chevalley Groups

The submodule structure of tyeyl modules for groups of type a1

The submodule structure of the so-called (Carter and Lusztig 131) weyl modules for groups of type A1 has been announced by Carter and Cline [2]. In their proof they use graph-theoretical machinery, and although it has not been published, we have been told by R. Carter that it is somewhat lengthy. In the present paper, we give a shorter proof using the concept of the hyperalgebra of a Lie algebra.

On homomorphisms between Weyl modules and Specht modules

The aim of this paper is to generalize a result of Carter and Lusztig(1) proving the existence of certain non-zero homomorphisms between Weyl modules for the special linear group SLn(K) over an infinite field of characteristic p, and between Specht modules for the symmetric group Sr over K.