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Anton Cox

Publications and source records attributed to Anton Cox.

10 recordsLinked to original sources
arXiv2023

Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras

We construct an explicit isomorphism between (truncations of) quiver Hecke algebras and Elias-Williamson's diagrammatic endomorphism algebras of Bott-Samelson bimodules. As a corollary, we deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated $p$-Kazhdan-Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. We hence give an elementary and more explicit proof of the main theorem of Riche-Williamson's recent monograph and extend their categorical equivalence to cyclotomic Hecke algebras, thus solving Libedinsky-Plaza's categorical blob conjecture.

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arXiv2010

Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra

We determine the decomposition numbers for the Brauer and walled Brauer algebra in characteristic zero in terms of certain polynomials associated to cap and curl diagrams (recovering a result of Martin in the Brauer case). We consider a second family of polynomials associated to such diagrams, and use these to determine projective resolutions of the standard modules. We then relate these two families of polynomials to Kazhdan-Lusztig theory via the work of Lascoux-Schützenberger and Boe, inspired by work of Brundan and Stroppel in the cap diagram case.

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arXiv2008

Alcove geometry and a translation principle for the Brauer algebra

There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra in characteristic zero. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra in characteristic zero can be reduced to a study of the block containing the weight 0. We define parabolic Kazhdan-Lusztig polynomials for the Brauer algebra and show in certain low rank examples that they determine standard module decomposition numbers and filtrations.

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arXiv2007

On the blocks of the walled Brauer algebra

We determine the blocks of the walled Brauer algebra in characteristic zero. These can be described in terms of orbits of the action of a Weyl group of type $A$ on a certain set of weights. In positive characteristic we give a linkage principle in terms of orbits of the corresponding affine Weyl group. We also classify the semisimple walled Brauer algebras in all characteristics.

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arXiv2006

A geometric characterisation of the blocks of the Brauer algebra

We give a geometric description of the blocks of the Brauer algebra $B_n(δ)$ in characteristic zero as orbits of the Weyl group of type $D_n$. We show how the corresponding affine Weyl group controls the representation theory of the Brauer algebra in positive characteristic, with orbits corresponding to unions of blocks.

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arXiv2005

Homomorphisms between Weyl modules for SL_3(k)

We classify all homomorphisms between Weyl modules for SL_3(k) when k is an algebraically closed field of characteristic at least three, and show that the Hom-spaces are all at most one-dimensional. As a corollary we obtain all homomorphisms between Specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three. We conclude by showing how a result of Fayers and Lyle on Hom-spaces for Specht modules is related to earlier work of Donkin for algebraic groups.

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arXiv2005

Homomorphisms and Higher Extensions for Schur algebras and symmetric groups

This paper surveys, and in some cases generalises, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined.

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arXiv2004

Representation theory of towers of recollement: theory, notes, and examples

We give an axiomatic framework for studying the representation theory of towers of algebras. We introduce a new class of algebras, contour algebras, generalising (and interpolating between) blob algebras and cyclotomic Temperley-Lieb algebras. We demonstrate the utility of our formalism by applying it to this class.

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