Abstract
This thesis presents a collection of results in the representation theory of the general linear group in defining characteristic, with a focus on multilinear constructions, explicit maps and combinatorial techniques.
We use tableaux to describe concrete models for the the Schur and Weyl endofunctors, and hence in particular the Weyl modules, their duals, and the Specht modules.
We establish a number of modular plethystic isomorphisms – isomorphisms between modules obtained by iterated Schur and Weyl endofunctors – for GL_{2}(K), where K is an arbitrary field. Our isomorphisms are generalisations of classical results, and require dualities that were not present in characteristic 0. An example is Hermite reciprocity, Sym_{m} Sym^{l} E isomorphic to Sym_{l} Sym^{m} E, where E is the natural representation. We exhibit explicit maps for our isomorphisms.
We study the image under the inverse Schur functor of the Specht module for the symmetric group, proving a necessary and sufficient condition on the indexing partition for this image to be isomorphic to the dual Weyl module in characteristic 2, and giving an elementary proof that this isomorphism holds in all cases in all other characteristics. We use this result to identify some indecomposable Specht modules. When the isomorphism does not hold, we describe some particular examples and prove some additional results, including a bound on the dimension of the kernel of the quotient map onto the dual Weyl module.
Finally we investigate a family of Markov chains on the set of simple representations of the finite group SL_{2}(F_{p}), defined by tensoring with a fixed simple module and choosing an indecomposable nonprojective summand. We draw connections between the properties of the chain and the representation theory of SL_{2}(F_{p}), emphasising symmetries of the tensor product. We also give a novel elementary proof of the decomposition of tensor products of simple modular SL_{2}(F_{p})representations.
We use tableaux to describe concrete models for the the Schur and Weyl endofunctors, and hence in particular the Weyl modules, their duals, and the Specht modules.
We establish a number of modular plethystic isomorphisms – isomorphisms between modules obtained by iterated Schur and Weyl endofunctors – for GL_{2}(K), where K is an arbitrary field. Our isomorphisms are generalisations of classical results, and require dualities that were not present in characteristic 0. An example is Hermite reciprocity, Sym_{m} Sym^{l} E isomorphic to Sym_{l} Sym^{m} E, where E is the natural representation. We exhibit explicit maps for our isomorphisms.
We study the image under the inverse Schur functor of the Specht module for the symmetric group, proving a necessary and sufficient condition on the indexing partition for this image to be isomorphic to the dual Weyl module in characteristic 2, and giving an elementary proof that this isomorphism holds in all cases in all other characteristics. We use this result to identify some indecomposable Specht modules. When the isomorphism does not hold, we describe some particular examples and prove some additional results, including a bound on the dimension of the kernel of the quotient map onto the dual Weyl module.
Finally we investigate a family of Markov chains on the set of simple representations of the finite group SL_{2}(F_{p}), defined by tensoring with a fixed simple module and choosing an indecomposable nonprojective summand. We draw connections between the properties of the chain and the representation theory of SL_{2}(F_{p}), emphasising symmetries of the tensor product. We also give a novel elementary proof of the decomposition of tensor products of simple modular SL_{2}(F_{p})representations.
Original language  English 

Qualification  Ph.D. 
Awarding Institution 

Supervisors/Advisors 

Publication status  Unpublished  2021 
Keywords
 general linear group
 symmetric group
 representation theory
 tableaux
 tabloids
 Schur endofunctor
 Weyl endofunctor
 Specht module
 Weyl module
 plethysm
 Schur functor