Decompositions of some Specht modules I

We give a decomposition as a direct sum of indecomposable modules of several types of Specht modules in characteristic $2$. These include the Specht modules labelled by hooks, whose decomposability was considered by Murphy. Since the main arguments are essentially no more difficult for Hecke algebras at parameter $q=-1$, we proceed in this level of generality.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1704.02413, arXiv:1704.0241

First Degree Cohomology of Specht Modules and Extensions of Symmetric Powers

Let $\Sigma_d$ denote the symmetric group of degree $d$ and let $K$ be a field of positive characteristic $p$. For $p>2$ we give an explicit description of the first cohomology group $H^1(\Sigma_d,{\rm{Sp}}(\lambda))$, of the Specht module ${\rm{Sp}}(\lambda)$ over $K$, labelled by a partition $\lambda$ of $d$. We also give a sufficient condition for the cohomology to be non-zero for $p=2$ and we find a lower bound for the dimension. Our method is to proceed by comparison with the cohomology for the general linear group $G(n)$ over $K$ and then to reduce to the calculation of ${\rm{Ext}}^1_{B(n)}(S^d E,K_\lambda)$, where $B(n)$ is a Borel subgroup of $G(n)$, $S^dE$ denotes the $d$th symmetric power of the natural module $E$ for $G(n)$ and $K_\lambda$ denotes the one dimensional $B(n)$-module with weight $\lambda$. The main new input is the description of module extensions by: extensions sequences, coherent triples of extension sequences and coherent multi-sequences of extension sequences, and the detailed calculation of the possibilities for such sequences. These sequences arise from the action of divided powers elements in the negative part of the hyperalgebra of $G(n)$.Comment: 94 page

On invariant ideals associated to classical groups

AbstractWe study the G1×G2-invariant ideals of the coordinate ring of the variety of nullforms, associated to a pair of classical groups over a field of characteristic zero

First degree cohomology of Specht modules and extensions of symmetric powers

Let Σ d denote the symmetric group of degree d and let K be a field of positive characteristic p. For p>2 we give an explicit description of the first cohomology group H 1(Σ d,Sp(λ)), of the Specht module Sp(λ) over K, labelled by a partition λ of d. We also give a sufficient condition for the cohomology to be non-zero for p=2 and we find a lower bound for the dimension. The cohomology of Specht modules has been considered in many papers including [10], [12], [15] and [21]. Our method is to proceed by comparison with the cohomology for the general linear group G=GL n(K) and then to reduce to the calculation of Ext B 1(S dE,K λ), where B is a Borel subgroup of G, where S dE denotes the dth symmetric power of the natural module E for G and K λ denotes the one dimensional B-module with weight λ. The main new input is the description of module extensions by: extensions sequences, coherent triples of extension sequences and coherent multi-sequences of extension sequences, and the detailed calculation of the possibilities for such sequences. These sequences arise from the action of divided powers elements in the negative part of the hyperalgebra of G. Our methods are valid also in the quantised context and we aim to treat this in a separate paper

On the Endomorphism Algebra of Specht Modules in Even Characteristic

Over fields of characteristic $2$, Specht modules may decompose and there is no upper bound for the dimension of their endomorphism algebra. A classification of the (in)decomposable Specht modules and a closed formula for the dimension of their endomorphism algebra remain two important open problems in the area. In this paper, we introduce a novel description of the endomorphism algebra of the Specht modules and provide infinite families of Specht modules with one-dimensional endomorphism algebra.Comment: 23 page