Test

Test

Nate Harman: Stability Phenomena in Representation Theory

28th August, 2023

For various natural sequences of groups, such as the general linear groups GLn or symmetric groups Sn, certain aspects of their representation theory act the same for all sufficiently large n. A classic example of this is Schur-Weyl duality, which gives a uniform description of degree d representations of GLn, provided n is at least d. I will discuss this and other examples of stability phenomena in representation theory, and how this sort of stabilization manifests itself in other areas of mathematics.

Ziqing Xiang: Quantum wreath product

28th July, 2023

The classical wreath product G ≀ Σd is a semidirect product Gd ⋊ Σd with Σd acting on Gd by permutations. We deform this classical wreath product by deforming G into an associative algebra B, deforming Σd into a Hecke algebra, and deforming the action. The result is called a quantum wreath product B ≀ H(d). Many variants of Hecke algebras can be viewed as quantum wreath products, hence could be treated in a unified manner.

In this talk, we will discuss necessary and sufficient conditions for quantum wreath products to have a basis of suitable size. We will also discuss some other structural results, the Schur algebras of these quantum wreath products, and their representations.

Greta Panova: Computational complexity in algebraic combinatorics II

8th April, 2023

Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives on their own and brought us some beautiful formulas and combinatorial interpretations.

The flagship hook-length formula counts the number of standard Young tableaux, which also give the dimension of the irreducible Specht modules of the symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory.

We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, which can formally explain the beauty we see and the difficulties we encounter in finding further formulas and "combinatorial interpretations". In the opposite direction, the 85 year old open problem on Kronecker coefficients of the symmetric group lead to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation-theoretic multiplicities in further detail, possibly asymptotically.

Greta Panova: Computational complexity in algebraic combinatorics I

8th April, 2023

Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives on their own and brought us some beautiful formulas and combinatorial interpretations.

The flagship hook-length formula counts the number of standard Young tableaux, which also give the dimension of the irreducible Specht modules of the symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory.

We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, which can formally explain the beauty we see and the difficulties we encounter in finding further formulas and "combinatorial interpretations". In the opposite direction, the 85 year old open problem on Kronecker coefficients of the symmetric group lead to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation-theoretic multiplicities in further detail, possibly asymptotically.

Duc-Khanh Nguyen: A generalization of the Murnaghan-Nakayama rule for K–k-Schur and k-Schur functions

21st March, 2023

We introduce a generalization of K-k-Schur functions and k-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of K-k-Schur functions and k-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for k-Schur functions, and explains it as a degeneration of the rule for K-k-Schur functions. In particular, many other special cases promise to be detailed in the future.

Sarah Peluse: Divisibility of character values of the symmetric group

9th February, 2023

In 2017, Miller computed the character tables of Sn for all n up to 38 and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and, based on his computations, Miller conjectured that almost all entries of the character table of Sn are divisible by any fixed prime power as n tends to infinity. In this talk, I will discuss joint work with K. Soundararajan that resolves this conjecture, and mention some related open problems.

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.

Nicolle González: Higher Rank Rational (q,t)-Catalan Polynomials and a Finite Shuffle Theorem

29th November, 2022

The classical shuffle theorem states that the Frobenius character of the space of diagonal harmonics is given by a certain combinatorial sum indexed by parking functions on square lattice paths. The rational shuffle theorem, conjectured by Gorsky-Negut and proven by Mellit, states that the geometric action on symmetric functions (described by Schiffmmann-Vasserot) of certain elliptic Hall algebra elements P(m,n) yield the bigraded Frobenius character of a certain Sn representation. This character is known as the Hikita polynomial. In this talk I will introduce the higher-rank rational (q,t)-Catalan polynomials and show these are equal to finite truncations of the Hikita polynomial. By generalizing results of Gorsky-Mazin-Vazirani and constructing an explicit bijection between rational semistandard parking functions and affine compositions, I will derive a finite analogue of the rational shuffle theorem in the context of spherical double affine Hecke algebras.

Giada Volpato: On the restriction of a character of Sn to a Sylow p-subgroup

15th November, 2022

The relevance of the McKay Conjecture in the representation theory of finite groups has led to investigate how irreducible characters decompose when restricted to Sylow p-subgroups. In this talk we will focus on the symmetric groups. Since the linear constituents of the restriction to a Sylow p-subgroup has been studied a lot by E. Giannelli and S. Law, we will concentrate on constituents of higher degree. In particular, we will describe the set of the irreducible characters which allow a constituent of a fixed degree, separating the cases of p being odd and p=2.

Loïc Poulain d’Andecy: KLR-type presentation of affine Hecke algebras of type B

1st November, 2022

KLR algebras of type A have been a revolution in the representation theory of Hecke algebras of a type A flavour, thanks to the the Brundan-Kleshchev-Rouquier isomorphism relating them explicitly to the affine Hecke algebra of type A. KLR algebras of other types exist but are not related to affine Hecke algebras of other types. In this talk I will present a generalisation of the KLR presentation for the affine Hecke algebra of type B and I will discuss some applications.