Tag - Schur-Weyl duality

Chun-Ju Lai: Quantum Wreath Products

29th May, 2024

A quantum wreath product is the algebra produced from a given (not necessarily commutative) algebra B, a positive integer d, and a choice of certain coefficients in B ⊗ B. Important examples include variants of the Hecke algebras, such as (1) affine Hecke algebras and their degenerate version, (2) Wan-Wang’s wreath Hecke algebras, (3) Kleshchev-Muth’s affinization algebras, (4) Rosso-Savage’s (affine) Frobenius Hecke algebras, (5) endomorphism algebras arising from Elias’s Hecke-type categories, (6) Mathas-Stroppel’s Rees affine Frobenius Hecke algebras, and (7) Hu algebra, which quantizes the wreath product Sm ≀ S2 between the symmetric groups. Our goal is to develop a uniform approach to the structure and representation theory in order to encompass known results which were proved in a case by case manner. In this talk, I’ll focus on the Schur-Weyl duality and the Clifford theory. Our theory is motivated by (and has application to) the Ginzburg-Guay-Opdam-Rouquier problem on quasi-hereditary covers of Hecke algebras for complex reflection groups.

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.

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.

Stephen Doty: Schur-Weyl duality for braid and twin groups via the Burau representation

15th February, 2022

The natural permutation representation of the symmetric group admits a q-analogue known as the Burau representation. The symmetric group admits two natural covering groups: the braid group of Artin and the twin group of Khovanov, obtained respectively by forgetting the cubic and quadratic relations in the Coxeter presentation of the symmetric group. By computing centralizers of tensor powers of the Burau representation, we obtain new instances of Schur-Weyl duality for braid groups and twin groups, in terms of the partial permutation and partial Brauer algebras. The method produces many representations of each group that can be understood combinatorially.

Samuel Creedon: Defining an Affine Partition Algebra

16th November, 2021

In this talk we motivate the construction of a new algebra called the affine partition algebra. We summarize some of its basic properties and describe an action which extends the Schur-Weyl duality between the symmetric group and partition algebra. We establish connections to the affine partition category defined recently by Brundan and Vargas and show that such a category is a full subcategory of the Heisenberg category.

Catharina Stroppel: Schur-Weyl dualities in non-semisimple cases

17th August, 2016

'Schur-Weyl duality' is often used to describe a concept in representation theory involving two kinds of symmetry that determine each other. In its original form it goes back to Schur and Weyl (around 1930) and describes an important interplay between the representation theory of the general linear and the symmetric group over the complex numbers. In this talk we will describe some generalizations of this phenomenon with a focus on modern, still open or recently solved questions. In particular we are interested in situations, where the involved algebras are not semisimple. We will indicate the origin of filtrations, homological properties and hidden gradings on the involved algebras and applications to the representation theory of Lie superalgebras.