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John Murray: A Schur-positivity conjecture inspired by the Alperin-Mckay conjecture

22nd March, 2022

The McKay conjecture asserts that a finite group has the same number of odd degree irreducible characters as the normalizer of a Sylow 2-subgroup. The Alperin-McKay (A-M) conjecture generalizes this to the height-zero characters in 2-blocks.

In his original paper, McKay already showed that his conjecture holds for the finite symmetric groups Sn. In 2016, Giannelli, Tent and the speaker established a canonical bijection realising A-M for Sn; the height-zero irreducible characters in a 2-block are naturally parametrized by tuples of hooks whose lengths are certain powers of 2, and this parametrization is compatible with restriction to an appropriate 2-local subgroup.

Now corresponding to a 2-block of the symmetric group Sn, there is a 2-block of a maximal Young subgroup of Sn of the same weight. An obvious question is whether our canonical bijection is compatible with restriction of height-zero characters between these blocks.

Attempting to prove this compatibility lead me to formulate a conjecture asserting the Schur-positivity of certain differences of skew-Schur functions. The corresponding skew-shapes have triangular inner-shape, but otherwise do not refer to the 2-modular theory. I will describe my conjecture and give positive evidence in its favour.

CJ Lai: Wreath products, Schur dualities, and quasi-hereditary covers

28th February, 2022

The Hecke algebra is in general not quasi-hereditary, meaning that its module category is not a highest weight category; while it admits a quasi-hereditary cover by the category đ’Ș of a certain rational Cherednik algebra due to Ginzburg-Guay-Opdam-Rouquier. It was later shown in type A that this category đ’Ș can be realized concretely as the module category of Dipper-James's q-Schur algebra, but this realization problem remains open beyond types A and B. An essential step for type D, i.e., the complex reflection group G(2,2,n), is to study Hu's Hecke subalgebra, which deforms a wreath product that is not a Coxeter group. In this talk, I'll introduce a new theory allowing us to take the wreath product of an algebra by a Hecke algebra. Before our work, wreath products related to Hecke algebras were worked out at the degenerate level by Wan-Wang. Our wreath product produces the Ariki-Koike algebras as special cases as well as new 'Hecke algebras' of wreath products between symmetric groups. These are the first steps towards answering the realization problem for complex reflection groups.

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.

George Seelinger: Diagonal harmonics and shuffle theorems

26th October, 2021

The Shuffle Theorem, conjectured by Haglund, Haiman, Loehr, Remmel and Ulyanov, and proved by Carlsson and Mellit, describes the characteristic of the Sn-module of diagonal harmonics as a weight generating function over labelled Dyck paths under a line with slope −1. The Shuffle Theorem has been generalized in many different directions, producing a number of theorems and conjectures. We provide a generalized shuffle theorem for paths under any line with negative slope using different methods from previous proofs of the Shuffle Theorem. In particular, our proof relies on showing a 'stable' shuffle theorem in the ring of virtual GLℓ-characters. Furthermore, we use our techniques to prove the Extended Delta Conjecture, yet another generalization of the original Shuffle Conjecture.

Diego Millan Berdasco: On the computation of decomposition numbers of the symmetric group

6th July, 2021

The most important open problem in the modular representation theory of the symmetric group is finding the multiplicity of the simple modules as composition factors of the Specht modules. In characteristic 0 the Specht modules are just the simple modules of the symmetric group algebra, but in positive characteristic they may no longer be simple. We will survey the rich interplay between representation theory and combinatorics of integer partitions, review a number of results in the literature which allow us to compute composition series for certain infinite families of Specht modules from a finite subset of them, and discuss the extension of these techniques to other Specht modules.

Max Gurevich: New constructions for irreducible representations in monoidal categories of type A

28th May, 2021

One ever-recurring goal of Lie theory is the quest for effective and elegant descriptions of collections of simple objects in categories of interest. A cornerstone feat achieved by Zelevinsky in that regard, was the combinatorial explication of the Langlands classification for smooth irreducible representations of p-adic GLn. It was a forerunner for an exploration of similar classifications for various categories of similar nature, such as modules over affine Hecke algebras or quantum affine algebras, to name a few. A next step - reaching an effective understanding of all reducible finite-length representations remains largely a difficult task throughout these settings.

Recently, joint with Erez Lapid, we have revisited the original Zelevinsky setting by suggesting a refined construction of all irreducible representations, with the hope of shedding light on standing decomposition problems. This construction applies the Robinson-Schensted-Knuth transform, while categorifying the determinantal Doubilet-Rota-Stein basis for matrix polynomial rings appearing in invariant theory. In this talk, I would like to introduce the new construction into the setting of modules over quiver Hecke (KLR) algebras. In type A, this category may be viewed as a quantization/gradation of the category of representations of p-adic groups. I will explain how adopting that point of view and exploiting recent developments in the subject (such as the normal sequence notion of Kashiwara-Kim) brings some conjectural properties of the RSK construction (back in the p-adic setting) into resolution. Time permits, I will discuss the relevance of the RSK construction to the representation theory of cyclotomic Hecke algebras.

Alexander Kleshchev: Irreducible restrictions from symmetric groups to subgroups

27th May, 2021

We motivate, discuss history of, and present a solution to the following problem: describe pairs (G, V) where V is an irreducible representation of the symmetric group Sn of dimension greater than 1 and G is a subgroup of Sn such that the restriction of V to G is irreducible. We do the same with the alternating group An in place of Sn.

Stacey Law: Sylow branching coefficients and a conjecture of Malle and Navarro

13th April, 2021

The relationship between the representation theory of a finite group and that of its Sylow subgroups is a key area of interest. For example, recent results of Malle–Navarro and Navarro–Tiep–Vallejo have shown that important structural properties of a finite group G are controlled by the permutation character 1P↑G, where P is a Sylow subgroup of G and 1P↑G denotes the trivial character of P. We introduce so-called Sylow branching coefficients for symmetric groups to describe multiplicities associated with these induced characters, and as an application confirm a prediction of Malle and Navarro from 2012, in joint work with E. Giannelli, J. Long and C. Vallejo.

Alexander Kleshchev: Irreducible restrictions from symmetric groups to subgroups

30th March, 2021

We motivate, discuss history of, and present a solution to the following problem: describe pairs (G,V) where V is an irreducible representation of the symmetric group Sn of dimension > 1 and G is a subgroup of Sn such that the restriction of V to G is irreducible. We do the same with the alternating group An in place of Sn.