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Stephen Donkin

Publications and source records attributed to Stephen Donkin.

16 recordsLinked to original sources
arXiv2026

On the Humphreys-Verma Conjecture for semisimple algebraic groups of rank $2$

Let $G$ be a connected, semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic. For each restricted dominant weight $\lambda$, there is the associated principal indecomposable $G_1$-module $Q_1(\lambda)$, where $G_1$ is the first infinitesimal subgroup of $G$. The assertion that, for every such $\lambda$, there exists a $G$-module whose restriction to $G_1$ is isomorphic to $Q_1(\lambda)$ is known as the Humphreys--Verma Conjecture. For groups of rank $2$, it was shown in \cite{BNPS1} that the Humphreys--Verma Conjecture holds in all cases except one, namely when $G$ is of type $G_2$, the characteristic is $2$, and $\lambda=0$. This case remained completely open. Moreover, in every previously resolved case, the module $Q_1(\lambda)$ could be realized as the restriction of a suitable tilting module. However, in \cite{BNPS2} it was shown that $Q_1(0)$ for $G_2$ in characteristic $2$ cannot arise as the restriction of a tilting module, thereby providing the first counterexample to a conjecture of the first author. In this paper, we construct a $G$-module whose restriction to $G_1$ is $Q_1(0)$, thereby establishing the Humphreys--Verma Conjecture in the last remaining rank $2$ case. Our construction provides the first known example of a $G$-structure on a principal indecomposable $G_1$-module that does not arise from a tilting module. This reveals a new phenomenon in the study of the Humphreys--Verma Conjecture and suggests new directions for understanding $G$-structures on principal indecomposable $G_1$-modules.

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arXiv2022

On the conjugation action for quantum general linear groups

We consider the conjugation action of a quantum group over an arbitrary field. In particular we consider the coordinate algebra of a quantised general linear group G(n), at an arbitrary nonzero parameter q, and give analogues of results of Kostant and Richardson

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arXiv2019

A Clebsch-Gordan decomposition in positive characteristic

Let $G$ be the special linear group of degree $2$ over an algebraically closed field $K$. Let $E$ be the natural module and $S^rE$ the $r$th symmetric power. We consider here, for $r,s\geq 0$, the tensor product of $S^rE$ and the dual of $S^sE$. In characteristic zero this tensor product decomposes according to the Clebsch-Gordan formula. We consider here the situation when $K$ is a field of positive characteristic. We show that each indecomposable component occurs with multiplicity one and identify which modules occur for given $r$ and $s$.

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arXiv2018

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.

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arXiv2017

Polynomially and Infinitesimally Injective Modules

The injective polynomial modules for a general linear group $G$ of degree $n$ are labelled by the partitions with at most $n$ parts. Working over an algebraically closed field of characteristic $p$, we consider the question of which partitions correspond to polynomially injective modules that are also injective as modules for the restricted enveloping algebra of the Lie algebra of $G$. The question is related to the "index of divisibility" of a polynomial module in general, and an explicit answer is given for $n=2$.

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arXiv2017

Composition Factors of Tensor Products of Symmetric Powers

We determine the composition factors of the tensor product $S(E)\otimes S(E)$ of two copies of the symmetric algebra of the natural module $E$ of a general linear group over an algebraically closed field of positive characteristic. Our main result may be regarded as a substantial generalisation of the tensor product theorem of Krop and Sullivan, on composition factors of $S(E)$. We earlier answered the question of which polynomially injective modules are infinitesimally injective in terms of the "divisibility index". We are now able to give an explicit description of the divisibility index for polynomial modules for general linear groups of degree at most $3$.

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arXiv2017

Invariants of Specht Modules

In [14] Hemmer conjectures that the module of fixed points for the symmetric group $Σ_m$ of a Specht module for $Σ_n$ (with $n>m$), over a field of positive characteristic $p$, has a Specht series, when viewed as a $Σ_{n-m}$-module. We provide a counterexample for each prime $p$. The examples have the same form for $p\geq 5$ and we treat the cases $p=3$ and $p=2$ separately.

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arXiv2017

Composition Factors of Tensor Products of Truncated Symmetric Powers

Let $G$ be the general linear group of degree $n$ over an algebraically closed field $K$ of characteristic $p>0$. We study the $m$-fold tensor product $\bar{S}(E)^{\otimes m}$ of the truncated symmetric algebra $\bar{S}(E)$ of the symmetric algebra $S(E)$ of the natural module $E$ for $G$. We are particularly interested in the set of partitions $λ$ occurring as the highest weight of a composition factor of $\bar{S}(E)^{\otimes m}$. We explain how the determination of these composition factors is related to the determination of the set of composition factors of the $m$-fold tensor product $S(E)^{\otimes m}$ of the symmetric algebra. We give a complete description of the composition factors of $\bar{S}(E)^{\otimes m}$ in terms of "distinguished" partitions. Our main interest is in the classical case, but since the quantised version is essentially no more difficult we express our results in the general context throughout.

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arXiv2017

Injective Schur Modules

We determine the partitions $λ$ for which the corresponding induced module (or Schur module in the language of Buchsbaum et. al., [1]) $\nabla(λ)$ is injective in the category of polynomial modules for a general linear group over an infinite field, equivalently which Weyl modules are projective polynomial modules. Since the problem is essentially no more difficult in the quantised case we address it at this level of generality. Expressing our results in terms of the representation theory of Hecke algebras at the parameter $q$ we determine the partitions $λ$ for which the corresponding Specht module is a Young module, when $1+q\neq 0$. In the classical case this problem was addressed by D. Hemmer, [12]. The nature of the set of partitions appearing in our solution gives a new formulation of Carter's condition on regular partitions. On the other hand, we note, in Remark 2.22, that the result on irreducible Weyl modules for the quantised Schur algebra $S_q(n,n)$, [17], Theorem 5.39, given in terms of Carter partitions, may be also used to obtain the main result presented here.

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arXiv2017

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 $H^1(Σ_d,{\rm{Sp}}(λ))$, of the Specht module ${\rm{Sp}}(λ)$ 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. 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 ${\rm{Ext}}^1_{B(n)}(S^d E,K_λ)$, 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 $K_λ$ denotes the one dimensional $B(n)$-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 $G(n)$.

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arXiv2013

Homological properties of quantised Borel-Schur algebras and resolutions of quantised Weyl modules

We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel-Schur subalgebras is deduced. This is used to prove the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the co-Specht modules for Hecke algebras.

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arXiv2009

The Brauer algebra and the symplectic Schur algebra

Let k be an algebraically closed field of characteristic p>0, let m,r be integers with m\ge1, r\ge0 and m\ge r and let S_0(2m,r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. Finally, we deduce from the previous results a new proof of the geometric description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.

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