Combinatorial representation theory is an extremely active area of research within pure mathematics with many connections to algebra, geometry and analysis. This 2009-10 RFG will examine applications of representation theory to subjects beyond these areas, with a particular interest in applying combinatorial methods.
Discrete Fourier analysis has been used by Diaconis to obtain rates of convergence for various random processes such as random walks on groups and recurrences for pseudo random number generation. In more recent papers, a similar analysis of the Metropolis algorithm turns out to involve Hecke algebras and the Jack symmetric polynomials that are special cases of Macdonald's symmetric polynomials. One goal of our investigation is to develop this connection between random processes and structures from algebraic combinatorics.
In this course, we will explore the monograph ``Group representations in probability and statistics'' by Diaconis as well as more recent papers. Students should have a strong knowledge of linear algebra, but we plan to briefly review basic group representation theory and Fourier analysis as part of the course. Students with a background in combinatorial representation theory may have more motivation for the investigations in the last part of the course.
In a series of substantial papers, Mulmuley and Sohoni have developed arguments termed ``Geometric Complexity Theory'' for attacking the famous $P \neq NP$ problem. Although their work is speculative, they relate positivity problems in representation theory to the existence of ``obstructions'' for polynomial-time algorithms in certain cases. One such positivity problem on the representation theory side has been solved using the Knutson and Tao Saturation Theorem.
While the program is far from complete and encompasses a great deal of material, we hope to review some of the key definitions and highlight a few of the main features of Geometric Complexity Theory. This reading group will present background material and research papers with a view towards understanding the Mulmuley--Sohoni work at a broad level.