ICTS

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.

Given a finite Abelian group we discuss how to count the number of flags of subgroups with specified orders using the theory of symmetric functions.

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.

In these lectures, I will provide an introduction to and overview of recent (2016-today) work on generating functions enumerating linear orbits and conjugacy classes of unipotent groups. We will mostly study these generating functions by means of their linearised siblings, so-called "ask zeta functions", obtained by averaging over sizes of kernels in collections of matrices. We will encounter several tools that have been employed and themes that emerged in the study of such zeta functions. In particular, we will explore both geometric and combinatorial ideas, as well as their interplay.

Permutation statistics have recently found several applications in asymptotic and enumerative algebra. The aim of this series of lectures is to develop some tools and intuition for uses of permutation statistics in this context. I will start by recalling some classical results, with pointers to recent developments in the world of zeta functions in algebra where these have found applications. I will then discuss a few extended examples of permutation statistics in the context of Coxeter and Weyl groups, and certain complex reflection groups.
These permutation statistics have applications to describing (numerators of) rational functions associated with various types of zeta functions, as well as to the enumeration of matrices over finite fields.

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.

In these lectures, I will provide an introduction to and overview of recent (2016-today) work on generating functions enumerating linear orbits and conjugacy classes of unipotent groups. We will mostly study these generating functions by means of their linearised siblings, so-called "ask zeta functions", obtained by averaging over sizes of kernels in collections of matrices. We will encounter several tools that have been employed and themes that emerged in the study of such zeta functions. In particular, we will explore both geometric and combinatorial ideas, as well as their interplay.

Permutation statistics have recently found several applications in asymptotic and enumerative algebra. The aim of this series of lectures is to develop some tools and intuition for uses of permutation statistics in this context. I will start by recalling some classical results, with pointers to recent developments in the world of zeta functions in algebra where these have found applications. I will then discuss a few extended examples of permutation statistics in the context of Coxeter and Weyl groups, and certain complex reflection groups.
These permutation statistics have applications to describing (numerators of) rational functions associated with various types of zeta functions, as well as to the enumeration of matrices over finite fields.

This lecture aims at introducing the general audience to a beautiful relationship between maths, physics and music. We will explore the evolution of ideas that led to the equal temperament, the 12-note system which prevails in much music worldwide. The main characters in this story are the wave equation in a string and its pure solutions, which are given by waves with integral periods; the relations between them which are rational numbers; and finally, a twist in the plot where equal-temperament is actually done with irrational numbers to achieve symmetry.

Webpage: https://www.icts.res.in/lectures/PLDec2024

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.

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 this talk, I will discuss zeta functions that count the number of twisted conjugacy classes of a fixed group.

Twisted conjugacy is a generalisation of the usual conjugacy, where we introduce a twist by an endomorphism. Specifically, given a group G and an automorphism f, we consider the action gx = gx f(g)^{-1}. The orbits of this action are known as twisted conjugacy classes, or Reidemeister classes.

Recent years have seen intensive investigation into the sizes of these classes. A major goal in this area is to classify groups where all classes are infinite. For groups that do not possess this property, the focus shifts to understanding the possible sizes of the classes, among all automorphisms.

In this talk, we will see that, as typical, these zeta functions admit Euler product decompositions with rational local factors, and we will explore how these zeta functions can be utilised to understand twisted conjugacy classes of certain nilpotent groups.

Let G be a nice (connected reductive) Lie group. An irreducible representation of G, when restricted to a maximal torus, decomposes into weights with multiplicity. We outline a procedure to compute symmetric polynomials (e.g., power sums) of this multiset of weights in terms of the highest weight. This is joint work with Rohit Joshi.

Similar to standard growth of (finitely generated) groups, one can define conjugacy growth of groups which, informally, counts the number of conjugacy classes in a ball of radius n in a Cayley graph. This was first studied by Riven for free groups, and techniques from geometry, combinatorics and formal language theory have proven to be useful for determining information about the conjugacy growth series for a variety of groups.
This talk will provide a survey on the key tools and results about conjugacy growth. Time permitting, I’ll also discuss joint work with Laura Ciobanu, where we studied conjugacy growth in dihedral Artin groups.

We study a variant of the Iwahori-Hecke algebra of a Coxeter group, whose generators T_i satisfy the braid relations but are assumed to be nilpotent (in parallel to Coxeter groups where the T_i are involutions, and 0-Hecke algebras where they are idempotent). Motivated by Coxeter (1957) and Broue-Malle-Rouquier (1998), we classify the finite-dimensional among these "generalized nil-Coxeter algebras". These turn out to be the usual nil-Coxeter algebras, and exactly one other type-A family of algebras, which have a finite "word basis" in the T_i and a unique longest word.

In the remaining time I will present joint work with Sutanay Bhattacharyya, in which we explore the "Temperley-Lieb" variant of the above, wherein all sufficiently long braid words are also killed. Now the finite-dimensional algebras obtained include ones with bases indexed by:
(a) words with a unique reduced expression (any Coxeter type),
(b) fully commutative words (counted by Stembridge),
(c) Catalan numbers (via the XYX algebras of Postnikov), and
(d) the \bar{12} avoiding signed permutations (in type B=C).