Seminars in Representations of Symmetric Groups

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Karlee Westrem: A new symmetric function identity with an application to symmetric group character values

7th October, 2024

Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of Σn character sums over a new set called Ev(λ). When investigating the alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of Σn characters over the set Ev(λ) equals 0.

Karin Erdmann: The Hemmer-Nakano Theorem and relative dominant dimension

30th May, 2024

Let ℋq(d) be the Iwahori-Hecke algebra of the symmetric group where q is a primitive ℓ-th root of unity, and let A = Sq(n,d) be the q-Schur algebra. Hemmer and Nakano proved amongst others that for ℓ ≥ 4, the Schur functor gives an equivalence between the category of A-modules with Weyl filtration, and the category of ℋq(d)-modules with dual Specht filtration, and that certain extension groups get identified. This has been a surprise and has inspired further research. In this talk we discuss some extensions of this result.

Jon Brundan: Isomeric Heisenberg and Kac–Moody categorification

29th May, 2024

The isomeric Heisenberg category acts naturally on a number of abelian categories appearing in the representation theory of the isomeric supergroup Q(n), and also on representations of Sergeev’s algebra which is related to the double covers of symmetric groups. I will explain an efficient way to convert an action of the isomeric Heisenberg category on these and other abelian categories into an action of a corresponding super Kac–Moody 2-category. To properly understand the odd simple root indexed by the element zero of the ground field requires the theory of odd symmetric functions developed by Ellis, Khovanov and Lauda, the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka, and the covering quantum groups defined and studied by Clark and Wang.

Weiqiang Wang: A new diagrammatic categorical setting for Schur dualities

29th May, 2024

The classical Schur duality is a simple yet powerful concept which relates the representations of the symmetric group and general linear Lie algebra, as well as combinatorics of symmetric functions. This admits a quantum deformation to a duality between a quantum group and Hecke algebra of type A. In this talk, we will describe several new simple diagrammatic (monoidal/quotient) categories, where old and new algebras behind (affine/cyclotomic) Schur duality emerge naturally. Our construction has new combinatorial implications on symmetric functions and RSK correspondence.

Christopher Drupieski: The Lie superalgebra of transpositions

28th May, 2024

In this talk I will report on work, joint with Jonathan Kujawa, to answer a series of questions originally posed by MathOverflow user WunderNatur in August 2022: Considering the group algebra ℂSn of the symmetric group as a superalgebra (by considering the even permutations in Sn to be of even superdegree and the odd permutations in Sn to be of odd superdegree), and then in turn considering ℂSn as a Lie superalgebra via the super commutator, what is the structure of ℂSn as a Lie superalgebra, and what is the structure of the Lie sub-superalgebra of ℂSn generated by the transpositions? The non-super versions of these questions were previously answered by Ivan Marin, with very different results. Time permitting, some thoughts on analogues of these questions for Weyl groups of types B/C and D may also be discussed.

Eoghan McDowell: Spin representations of the symmetric group which reduce modulo 2 to Specht modules

16th April, 2024

When do two ordinary irreducible representations of a group have the same p-modular reduction? In this talk I will address this question for the double cover of the symmetric group, and more generally give a necessary and sufficient condition for a spin representation of the symmetric group to reduce modulo 2 to a multiple of a Specht module (in the sense of Brauer characters or in the Grothendieck group). I will explain some of the techniques used in the proof, including describing a function which swaps adjacent runners in an abacus display for the labelling partition of a character.

Alexander Wilson: Super Multiset RSK and a Mixed Multiset Partition Algebra

22nd January, 2024

Through dualities on representations on tensor powers and symmetric powers respectively, the partition algebra and multiset partition algebra have been used to study long-standing questions in the representation theory of the symmetric group. These algebras enjoy distinguished bases whose product can be described on graph-theoretic diagrams. We extend this story to exterior powers, leading to the introduction of the mixed multiset partition algebra and a generalization of RSK that links the algebra’s graph-theoretic basis to a tableau basis for its irreducible representations.

Eric Marberg: From Klyachko models to perfect models

5th December, 2023

In this talk a "model" of a finite group or semisimple algebra means a representation containing a unique irreducible subrepresentation from each isomorphism class. In the 1980s Klyachko identified an elegant model for the general linear group over a finite field with q elements. There is an informal sense in which taking the q→1 limit of Klyachko's construction gives a model for the symmetric group, which can be extended to its Iwahori-Hecke algebra. The resulting Hecke algebra representation is a special case of a "perfect model", which is a more flexible construction that can be considered for any finite Coxeter group. In this talk, I will classify exactly which Coxeter groups have perfect models, and discuss some notable features of this classification. For example, each perfect model gives rise to a pair of related W-graphs, which are dual in types B and D but not in type A. Various interesting questions about these W-graphs remain open.

Alexander Yong: Newell-Littlewood numbers

28th November, 2023

The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question: Which multiplicities are non-zero? In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's non-vanishing results for the Newell-Littlewood numbers.

Geordie Williamson: A Panoramic View of Modular Representation Theory

6th October, 2023

I will try to give a glimpse of exciting developments in representation theory over the last two decades. A central focus will be on the representations of symmetric groups over the complex numbers and fields of positive characteristic. Over the complex numbers our understanding is very good, however the case of positive characteristic fields has turned out to be more complicated than (I suspect) the pioneers would have ever imagined. Remarkably, there appears to be a way forward which combines ideas which emerged in the Langlands programme with techniques from mod p algebraic topology (Smith theory).