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Let F be a non-Archimedean local field F with ring of integers O and a finite residue field k of characteristic greater than three. While the representations of finite groups of Lie type GL_n(k) and of the p-adic groups GL_n(F) are well studied, the representations of GL_n(O) remain far less understood.

In this talk, we will explore the challenges involved in constructing the complex irreducible representations of GL_n(O), highlighting key differences from the case of GL_n(k). We will then present a method for constructing irreducible representations of GL_3(O). This is based on a recent joint work with Uri Onn and Amritanshu Prasad.

In the symmetric group S_n, there are n^{n-2} ways to write each n-cycle as a product of the minimum number of transpositions. This theorem has numerous extensions: in the symmetric group, such questions are tied to the enumeration of embedded maps on surfaces and moduli spaces of curves, while in real and complex reflection groups the analogous theorem is one ingredient in the Catalan--Coxeter theory and the study of the lattice of W-noncrossing partitions.

About a decade ago, with Vic Reiner and Dennis Stanton, we studied the analogue of this result for the general linear group over a finite field F_q. In this setting, the role of the n-cycle is taken by a Singer cycle, and that of the transpositions by the reflections; we showed that the number of factorizations is (q^n - 1)^{n - 1}. In this talk, I will discuss ongoing work, joint with C. Ryan Vinroot, that extends this work to a larger family of linear and unitary groups over a finite field.

This is an expository talk on some combinatorial aspects of matrix groups over rings. We first discuss results on different types of bounded factorizations of elementary Chevalley groups over certain commutative rings. These rings include local rings, finite fields, rings of matrix-valued holomorphic maps on Stein spaces as well as number rings. The results are intimately related to deep properties such as the congruence subgroup property and Kazhdan’s property T among other things. We also briefly mention growth functions which arise naturally, whose analytic information encodes group theoretic information.

Over the three lectures and the tutorial, I want to explore some problems in enumerative algebra that can be understood by associating lattices to semistandard Young tableaux. I will introduce a family of rational functions called Hall--Littlewood--Schubert series (HLS series), which was recently introduced by Christopher Voll and myself. The enumerative problems we discuss will be solved by judicious substitutions of the variables of the HLS series.

Over the three lectures and the tutorial, I want to explore some problems in enumerative algebra that can be understood by associating lattices to semistandard Young tableaux. I will introduce a family of rational functions called Hall--Littlewood--Schubert series (HLS series), which was recently introduced by Christopher Voll and myself. The enumerative problems we discuss will be solved by judicious substitutions of the variables of the HLS series.

Over the three lectures and the tutorial, I want to explore some problems in enumerative algebra that can be understood by associating lattices to semistandard Young tableaux. I will introduce a family of rational functions called Hall--Littlewood--Schubert series (HLS series), which was recently introduced by Christopher Voll and myself. The enumerative problems we discuss will be solved by judicious substitutions of the variables of the HLS series.

The series of three short lectures provides an introduction to the subject of subgroup growth and related directions of research in asymptotic group theory. For instance, we will discuss polynomial subgroup growth and look at subgroup zeta functions of finitely generated nilpotent groups. We also cover methods from the theory of compact p-adic Lie groups, which have applications to subgroup growth and representation growth.

The series of three short lectures provides an introduction to the subject of subgroup growth and related directions of research in asymptotic group theory. For instance, we will discuss polynomial subgroup growth and look at subgroup zeta functions of finitely generated nilpotent groups. We also cover methods from the theory of compact p-adic Lie groups, which have applications to subgroup growth and representation growth.

The series of three short lectures provides an introduction to the subject of subgroup growth and related directions of research in asymptotic group theory. For instance, we will discuss polynomial subgroup growth and look at subgroup zeta functions of finitely generated nilpotent groups. We also cover methods from the theory of compact p-adic Lie groups, which have applications to subgroup growth and representation growth.