Combinatorial Representation Theory

Dartmouth College, Hanover NH

Investigators

Abstract

Algebraic objects are often perceived as complicated and mysterious. In combinatorial representation theory one seeks simpler ways to represent them in order to understand, organize and apply them to other subjects. This project deals with tensor products of representations, which is a way to combine representations. The main problem that the project seeks to understand is how to decompose tensor products into sums of simpler representations. One can think about it as the problem of recovering individual signals from a mixture of signals. The tensor product decomposition is a difficult problem but also very important, because it shows up in many fields, including algebraic combinatorics, complexity theory, and statistics, and has applications in medicine, computer vision, physics, chemistry, and fast matrix multiplication. A fundamental open problem in combinatorial representation theory is to describe in the decomposition of the tensor product the multiplicities of two irreducible representations of the symmetric group; these multiplicities are called Kronecker coefficients. The problem of finding a combinatorial interpretation for these coefficients has been labeled as one of the "main problems in combinatorial representation theory." The PI and collaborators have been able to connect the Kronecker coefficients to the partition algebra. This connection led to the discovery of the "universal characters" of the symmetric group, which are symmetric functions that specialize to characters of the symmetric group when evaluated at roots of unity. These symmetric functions connect several difficult problems in combinatorial representation theory and will lead to the resolution of several of these problems.