Combinatorial Representation Theory

In our paper, we introduce a new method for estimating incidences via representation theory. We obtain several applications to various sums with multiplicative characters and to Zaremba's conjecture from number theory.

Young diagrams are ubiquitous in combinatorics and representation theory. Here we explain these diagrams, focusing on how they are used to classify representations of the symmetric groups $S_n$ and various "classical groups": famous groups of matrices such as the general linear group $\mathrm{GL}(n,\mathbb{C})$ consisting of all invertible $n \times n$ complex matrices, the special linear group $\mathrm{SL}(n,\mathbb{C})$ consisting of all $n \times n$ complex matrices with d...

Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Computer Science, and the rapid development of the latter. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern ...

This paper is a survey of some of the ways in which the representation theory of the symmetric group has been used in voting theory and game theory. In particular, we use permutation representations that arise from the action of the symmetric group on tabloids to describe, for example, a surprising relationship between the Borda count and Kemeny rule in voting. We also explain a powerful representation-theoretic approach to working with linear symmetric solution concepts in c...

This survey contains a recollection of results, problems and conversations which go back to the early years of Representation Theory and Tilting Theory.

We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making systematic use of finite groupoids. This provides a road map for the various approaches to the axiomatic representation theory of finite groups, as well as some details which are hard to find in writing.

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for student research, it is hopefully also useful for a wider audience. It is the author's intention to keep the material current, thus all feedback and updates are greatly appreciated.

We adapt methods from quiver representation theory and Hall algebra techniques to the counting of representations of virtually free groups over finite fields. This gives rise to the computation of the E-polynomials of $\mathbf{GL}_d(\mathbb{C})$-character varieties of virtually free groups. As examples we discuss the representation theory of $\mathbb{D}_\infty$ , $\mathbf{PSL}_2(\mathbb{Z})$ , $\mathbf{SL}_2(\mathbb{Z})$ , $\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\math...

An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is investigated in detail. The resulting representations are completely classified and include the irreducible ones.

This note being devoted to some aspects of the inverse problem of representation theory contains a new insight into it illustrated by two topics. The attention is concentrated on the manner of representation of abstract objects by the concrete ones as well as on the abstract objects themselves. The results of researches allow to state that the actual richness and attractiveness of the inverse problem of representation theory are based not only on a large scope of various inte...