The vertices and sources of the basic spin module for the symmetric group in characteristic 2

In this article we determine the vertices and sources of the basic spin module for the symmetric group S_n of degree n over a field of characteristic 2.     

Green vertices and sources of simple modules of the symmetric group labelled by hook partitions

We determine the Green vertices and sources of many of the simple modular representations of the finite symmetric group being parametrized by hook partitions. Received: 20 August 2006     

Hooks generate the representation ring of the symmetric group

We prove that the representation ring of the symmetric group on n$n$ letters is generated by the exterior powers of its natural (n−1)$(n-1)$-dimensional representation. The proof we give illustrates a strikingly simple formula due to Dvir. We provide an application and investigate a possible generalization of this result to some other reflection groups.     

Two theorems on the vertices of Specht modules

Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module $S^\lambda$ has non-trivial cyclic vertex if and only if $\lambda$ has p-weight 1. 2. If p does not divide n and $S^{(n-r, 1^r)}$ is indecomposable then its vertex is a p-Sylow subgroup of $S_{n-r-1} \times S_r$.Received: 15 August 2002     

The ordinary quiver of a weight three block of the symmetric group is bipartite

Suppose F is a field of characteristic p?5, and that B is a p-block of the symmetric group Sn of defect 3. We prove that the Ext¹-quiver of B is bipartite, with the bipartition being described in a simple way using the leg-lengths of p-hooks of partitions.     

Minors of symmetric and exterior powers

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory.     

Spin invariant theory for the symmetric group

We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur Q-functions and a shifted q-hook formula. In addition, we provide a bijective proof for a formula of the principal specialization of the Schur Q-functions.     

Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.     

Irreducible tensor products for alternating groups in characteristic 2

In this paper we completely characterise irreducible tensor products of a basic spin module with an irreducible module for alternating groups in characteristic 2. This completes the classification of irreducible tensor products of representations of alternating groups.     

On the asymptotic properties of the Rees powers of a module

Let R be a commutative ring and G a free R-module with finite rank e>0. For any R-submodule E⊂G one may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components E_n, n≥0, of the image, that we shall call the n-th Rees powers of E (with respect to the embedding E⊂G). In this work we prove some asymptotic properties of the R-modules E_n, n≥0, which extend well known similar ones for the case of ideals, among them Burch’s inequality for the analytic spread.     

Large dimension homomorphism spaces between Specht modules for symmetric groups

Let F be a field of characteristic p. We show that Hom_FΣn(S^λ,S^μ) can have arbitrarily large dimension as n and p grow, where S^λ and S^μ are Specht modules for the symmetric group Σ_n. Similar results hold for the Weyl modules of the general linear group. Every previously computed example has been at most one-dimensional, with the exception of Specht modules over a field of characteristic two. The proof uses the work of Chuang and Tan, providing detailed information about the radical series of Weyl modules in Rouquier blocks.     

Generalized blocks for symmetric groups

We study, via character-theoretic methods, an ℓ-analogue of the modular representation theory of the symmetric group, for an arbitrary integer ℓ≥2. We find that many of the invariants of the usual block theory (ie. when ℓ is prime) generalize in a natural fashion to this new context. Oblatum 21-III-2002 & 5-VIII-2002?Published online: 8 November 2002     

Regular orbits on the deleted permutation module

Let S _d denote the symmetric group of degree d. Let F be the field of r elements, and let S(d, r) = S _d × F*. Let V(d, r) be the deleted permutation module of dimension d – 1 over F, viewed as an S(d, r)-module. We determine the pairs (d, r) such that S(d, r) has a regular orbit on V (d, r). This question arose from D. Goodwin’s work on the quasisimple case of the k(GV)-problem. Received: 17 April 2007     

Subconstituents of orthogonal graphs of odd characteristic

In this paper, the subconstituents of the isotropic orthogonal graphs over finite fields of odd characteristic are studied. The first subconstituent is strongly regular, while the second subconstituent is edge-regular. The full automorphism groups of these two subconstituents have also been determined.     

Action of generalized symmetric groups on symmetric and exterior powers of their natural representations

We establish upper and lower bounds on the dimension of the space spanned by the symmetric powers of the natural character of generalized symmetric groups. We adapt the methods of Savitt and Stanley from [4 Savitt, D., Stanley, R. P. (2000). A note on the symmetric powers of the standard representation of S_n. Electron. J. Combin. 7:R6. [Google Scholar]] to obtain bounds both over the complex numbers and in prime characteristic.