Characteristic-free resolutions of Weyl and Specht modules

We construct explicit resolutions of Weyl modules by divided powers and of co-Specht modules by permutational modules. We also prove a conjecture by Boltje and Hartmann (2010) [7] on resolutions of co-Specht modules.     

Reducible Specht modules

James and Mathas conjecture a criterion for the Specht module S^λ for the symmetric group to be irreducible over a field of prime characteristic. We extend a result of Lyle to prove this conjecture in one direction; our techniques are elementary.     

Homomorphisms between Specht modules

In positive characteristic, the Specht modules S corresponding to partitions are not necessarily irreducible, and understanding their structure is vital to understanding the representation theory of the symmetric group. In this paper, we address the related problem of finding the spaces of homomorphisms between Specht modules. Indeed in [2], Carter and Payne showed that the space of homomorphisms from S to S is non-zero for certain pairs of partitions and where the Young diagram for is obtained from that for by moving several nodes from one row to another. We also consider partitions of this type, and, by explicitly examining certain combinations of semi-standard homomorphisms, we are able to give a constructive proof of the Carter–Payne theorem and to generalise it.Mathematics Subject Classification (2000): 20C30     

Specht modules with abelian vertices

In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily p ²-cores where p is the characteristic of the underlying field. Furthermore, in the case of p≥3, or p=2 and μ is 2-regular, we show that the complexity of the Specht module S ^μ is precisely the p-weight of the partition μ. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module S^(pp)S^{(p^{p})} for p≥3.     

Specht modules and chromatic polynomials

An explicit formula for the chromatic polynomials of certain families of graphs, called bracelets', is obtained. The terms correspond to irreducible representations of symmetric groups. The theory is developed using the standard bases for the Specht modules of representation theory, and leads to an effective means of calculation.     

A simplified presentation of Specht modules

Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our presentation applies to all Specht modules, and is of a similar nature to a recent result by Friedmann-Hanlon-Stanley-Wachs that applies to staircase partitions. We show that our presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation.     

A categorification of integral Specht modules

We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category for the Lie algebra .

     

Restricting Specht modules of cyclotomic Hecke algebras

This paper proves that the restriction of a Specht module for a (degenerate or non-degenerate) cyclotomic Hecke algebra, or KLR (Khovanov-Lauda-Rouquier) algebra, of type A has a Specht filtration.     

Permutation modules and I-cohomology

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The complexity of certain Specht modules for the symmetric group

During the 2004–2005 academic year the VIGRE Algebra Research Group at the University of Georgia (UGA VIGRE) computed the complexities of certain Specht modules S ^λ for the symmetric group Σ _d , using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The UGA VIGRE Algebra Group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of λ is built out of p×p blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the p-weight of partitions and branching.     

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.     

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     

Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Suppose \(\lambda \) and \(\mu \) are integer partitions with \(\lambda \supseteq \mu \). Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram \(\lambda \backslash \mu \), which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting cover-inclusive Dyck tilings. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call cover-expansive Dyck tilings. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson. We then discuss the connections with recent results of Kim et al., who give a simple expression for the sum, over all \(\mu \), of the number of cover-inclusive Dyck tilings of \(\lambda \backslash \mu \). Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the homogeneous Garnir relations for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras.     

Permutation modules of profinite groups

Given an arbitrary profinite group G and a commutative domain R, we define the notion of permutation RG-module which generalizes the known notion from the representation theory of profinite groups. We establish an independence theorem of such a module as an R-module over a ring of scalars.     

Free resolutions for differential modules over differential algebras

A free resolution (R, d + h) → (M, d) for a DG-module (M, d) over a DG-algebra (A, d) is constructed in the sense of a perturbation of the differential in a free bigraded resolution (R, d) → M of the underlying graded module M over an underlying graded algebra A. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.