Representations of the general linear group with multilinear constructions. / McDowell, Eoghan.

2021. 216 p.

Research output: ThesisDoctoral Thesis

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@phdthesis{a0e62a32cf4646f69da1197f36548a62,
title = "Representations of the general linear group with multilinear constructions",
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 GL2(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, Symm Syml E isomorphic to Syml Symm 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 SL2(Fp), defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We draw connections between the properties of the chain and the representation theory of SL2(Fp), emphasising symmetries of the tensor product. We also give a novel elementary proof of the decomposition of tensor products of simple modular SL2(Fp)-representations.",
keywords = "general linear group, symmetric group, representation theory, tableaux, tabloids, Schur endofunctor, Weyl endofunctor, Specht module, Weyl module, plethysm, Schur functor",
author = "Eoghan McDowell",
year = "2021",
language = "English",
school = "Royal Holloway, University of London",

}

RIS

TY - THES

T1 - Representations of the general linear group with multilinear constructions

AU - McDowell, Eoghan

PY - 2021

Y1 - 2021

N2 - 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 GL2(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, Symm Syml E isomorphic to Syml Symm 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 SL2(Fp), defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We draw connections between the properties of the chain and the representation theory of SL2(Fp), emphasising symmetries of the tensor product. We also give a novel elementary proof of the decomposition of tensor products of simple modular SL2(Fp)-representations.

AB - 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 GL2(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, Symm Syml E isomorphic to Syml Symm 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 SL2(Fp), defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We draw connections between the properties of the chain and the representation theory of SL2(Fp), emphasising symmetries of the tensor product. We also give a novel elementary proof of the decomposition of tensor products of simple modular SL2(Fp)-representations.

KW - general linear group

KW - symmetric group

KW - representation theory

KW - tableaux

KW - tabloids

KW - Schur endofunctor

KW - Weyl endofunctor

KW - Specht module

KW - Weyl module

KW - plethysm

KW - Schur functor

M3 - Doctoral Thesis

ER -