On the blocks of semisimple algebraic groups and associated generalized Schur algebras

In this paper we give a new proof for the description of the blocks of any semisimple simply connected algebraic group when the characteristic of the field is greater than 5. The first proof was given by Donkin and works in arbitrary characteristic. Our new proof has two advantages. First we obtain a bound on the length of a minimum chain linking two weights in the same block. Second we obtain a sufficient condition on saturated subsets π of the set of dominant weights which ensures that the blocks of the associated generalized Schur algebra are simply the intersection of the blocks of the algebraic group with the set π. However, we show that this is not the case in general for the symplectic Schur algebras, disproving a conjecture of Renner

A note on the tensor product of restricted simple modules for algebraic groups

Let G be a semisimple simply connected algebraic group over an algebraically closed field of positive characteristic p. Denote by G1 its first Frobenius kernel. In this note, we determine for which group G the restriction to G1 of any indecomposable G-summand of the tensor product of any two restricted simple G-modules remains indecomposable

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

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