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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.

Michael Larsen: Character estimates for classical finite simple groups

27th May, 2021

This is intended to complement the recent talk of Pham Huu Tiep in this seminar but will not assume familiarity with that talk. The estimates in the title are upper bounds of the form |χ(g)|≤χ(1)α, where χ is irreducible and α depends on the size of the centralizer of g. I will briefly discuss geometric applications of such bounds, explain how probability theory can be used to reduce to the case of elements g of small centralizer, discuss the level theory of characters, and conclude with the reduction to the case of characters χ of large degree. For such pairs (g,χ), exponential character bounds are trivial.

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.

Jon Carlson: The endomorphism ring of the trivial module

19th April, 2021

Let k be a field of characteristic p and let H be a finite group or group scheme. We show that the negative Tate cohomology ring Ĥ≤0(H,k) can be realized as the endomorphism ring of the trivial module in a Verdier localization of the stable category of kG-modules for G an extension of H. This means in some cases that the endomorphism of the trivial module is a local ring with infinitely generated radical with square zero. This stands in stark contrast to some known calculations in which the endomorphism ring of the trivial module is the degree zero component of a localization of the cohomology ring of the group.

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.

Pham Huu Tiep: Character bounds for finite simple groups

12th April, 2021

Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We first discuss some such applications. Then we will report on recent results that produce such character bounds.

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.

Lucas Mason-Brown: What is a Unipotent Representation?

22nd February, 2021

The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(𝔽q) be the group of 𝔽q-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(𝔽q). He showed:

   1.  All irreducible representations of G(𝔽q) can be constructed from a finite set of building blocks -- called 'unipotent representations.'
   2.  Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(𝔽q).

Now, replace 𝔽q by ℂ, the field of complex numbers, and replace G(𝔽q) with G(ℂ). There is a striking analogy between the finite-dimensional representation theory of G(𝔽q) and the unitary representation theory of G(ℂ). This analogy suggests that all unitary representations of G(ℂ) can be constructed from a finite set of building blocks - called 'unipotent representations' - and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of 'special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.

Hung Nguyen: Group characters and their fields of values

9th November, 2020

Representations and characters have been proved to be very useful tools to study finite groups. I will discuss some results on the relationship between fields of values of characters of a finite group and the structure of the group, from a classical result of Burnside on real-valued characters and Navarro-Tiep's result on rational-valued characters of odd degree to very recent results of my collaborators and myself on (almost) p-rational characters.