Lots of real world problems lead to group valued data...500 people ranking 5 types of chocolate chip cookies gives data on S(5). Daily birth rate data in New York City to data on C(365). There is even monster valued data- does the product replacement algorithm algorithm generate random looking elements. Fourier analysis on the groups calls for detailed knowledge of both characters and representations. It needs non-commutative analogs of the FFT. I will try to explain the statistics to people who know about groups (but not statistics). Again, there are many open problems

A character ratio for a finite group G is a complex number of the form chi(g)/chi(1), where chi is an irreducible character of G. I shall discuss some recent results on character ratios for finite groups of Lie type, I shall also mention several diverse applications, to random walks, probabilistic generation, and representation varieties of some classes of infinite groups

Character methods, probabilistic methods, and their combination, have been instrumental in the solution of various problems in group theory over the years.

In the talk I will provide a brief background and then focus on some brand new results and conjectures, related to random walks and to word maps.

In particular I will discuss a solution (joint work with Larsen and Tiep) of the so-called Probabilistic Waring Problem for finite simple groups, which applies strong new character bounds.

We give an executive summary of recent work with Ivan Losev, which connects modular representation theory in type A with a diagrammatic category of singular Soergel bimodules. As a consequence, various multiplicities in modular representation theory are encoded by p-Kazhdan-Lusztig polynomials. All this is in spite of the observation that the two sides of this story categorify different objects: one categorifies Fock space, while the other a space spanned by virtual partitions. We explain a magic trick due to Losev which allows one to compare this different categorical representations.

We will explain how double affine structures and two-variable polynomials arise from the modular representation theory of finite groups of Lie type in non-describing characteristic

One of the few tools one has when studying the characters of a finite group $G$ is given by induction $\mathrm{Ind}_H^G$ from a subgroup $H \leqslant G$. If $G$ is a finite reductive group and $L \leqslant G$ is a Levi subgroup then, in 1976, Deligne and Lusztig have defined a geometric analogue of this construction, which is typically referred to as Deligne—Lusztig induction $R_L^G$. A fundamental problem is to understand the decomposition of $R_L^G(\chi)$ into its irreducible constituents for any irreducible character $\chi$ of $L$. If $L$ is a torus then this problem was completely settled by Lusztig in the mid to late 80s. Using Shintani descent Asai solved this problem when $\chi$ is a unipotent character of $L$. Assuming $G$ comes from an algebraic group with connected centre then Shoji obtained generalisations of Asai’s results for arbitrary characters. In this talk we describe a new approach to this problem which relies on the validity of the Mackey formula

The McKay conjecture was refined in 2004 by G. Navarro 
taking into account the action of Galois automorphisms on characters. 
This refinement is known as the Galois McKay conjecture, and it is one 
of the last counting conjectures without a reduction theorem. In this 
talk I will present a possible reduction theorem for the Galois McKay 
conjecture to a question on finite simple groups. This is joint work 
with G. Navarro and B. Späth.

 

[This is a work in progress with Bonnafé-Broué-Malle-Michel-Rouquier.] In this talk I will explain how to construct a non-trivial action of SL2(Z) on the vector space spanned by unipotent characters. The existence of such an action follows from Lusztig's classification of unipotent characters, but I will present a global construction relying on traces of automorphisms of Deligne--Lusztig varieties. The advantage of this approach is that it can be naturally generalized to Spetses, thus giving candidates for Fourier matrices and unipotent character sheaves for Spetses.