Tag - Symmetric functions

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.

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.

Arun Ram: Lusztig varieties and Macdonald polynomials

27th May, 2024

In type A, the Macdonald polynomials and the integral from Macdonald polynomials are related by a plethystic transformation. We interpret this plethystic transformation geometrically as a relationship between nilpotent parabolic Springer fibres and nilpotent Lusztig varieties. This points the way to a generalization of modified Macdonald polynomials and integral form Macdonald polynomials to all Lie types. But these generalizations are not polynomials, they are elements of the Iwahori-Hecke algebra of the finite Weyl group. This work concerns the generalization of, and connection between, a 1997 paper of Halverson-Ram (which counts points of nilpotent Lusztig varieties over a finite field) and a 2017 paper of Mellit (which counts points of nilpotent parabolic affine Springer fibres over a finite field).

Pavel Turek: On stable modular plethysms of the natural module of SL2(𝔽p) in characteristic p

24th January, 2023

To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL2(𝔽p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL2(𝔽p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL2(𝔽p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are 'close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.

Rowena Paget: Plethysm and the Partition Algebra

10th January, 2023

The symmetric group Smn acts naturally on the collection of set partitions of a set of size mn into n sets each of size m. The irreducible constituents of the associated ordinary character are largely unknown; in particular, they are the subject of the longstanding Foulkes Conjecture. There are equivalent reformulations using polynomial representations of infinite general linear groups or using plethysms of symmetric functions. I will review plethysm from these three perspectives before presenting a new approach to studying plethysm: using the Schur-Weyl duality between the symmetric group and the partition algebra. This method allows us to study stability properties of certain plethysm coefficients. This is joint work with Chris Bowman. If time permits, I will also discuss some new results with Chris Bowman and Mark Wildon.

Mark Wildon: Plethysms, polynomial representations of linear groups and Hermite reciprocity over an arbitrary field

27th April, 2021

Let E be a 2-dimensional vector space. Over the complex numbers the irreducible polynomial representations of the special linear group SL(E) are the symmetric powers SymrE. Composing polynomial representations, for example to form Sym4 Sym2E, corresponds to the plethysm product on symmetric functions. Expressing such a plethysm as a linear combination of Schur functions has been identified by Richard Stanley as one of the fundamental open problems in algebraic combinatorics. In my talk I will use symmetric functions to prove some classical isomorphisms, such as Hermite reciprocity Symm SymrE ≅ SymrSymmE, and some others discovered only recently in joint work with Rowena Paget. I will then give an overview of new results showing that, provided suitable dualities are introduced, Hermite reciprocity holds over arbitrary fields; certain other isomorphisms (we can prove) have no modular generalization.