Combinatorics and Asymptotics of Structure Constants from Representation Theory and Algebra

University Of Southern California, Los Angeles CA

Investigators

Abstract

Symmetries and patterns capture the essence of our complex world and give the abstraction necessary to study it with algebraic and discrete methods. As objects change and interact, so do the inherent symmetries. How do symmetries combine, restrict, project or transform into other symmetries? How do we decompose a complex system of irreducible components? In general we want to describe this interaction quantitatively -- how many components of each type are contained in another bigger symmetry structure. While the computational complexity of the problem in general hints that no "closed-form" answer would exist, it is the goal of this project to find these numbers approximately and see how a typical structure looks like. These problems appear in many disguises, and lie at the intersection of combinatorics, algebra, representation theory, probability and statistical mechanics, and computational complexity theory. More precisely, the PI aims to solve problems in algebraic combinatorics involving "structure constants" and Young tableaux. Structure constants are generally defined as the multiplicities of irreducible symmetric or general linear group representations in the decomposition of tensor products or compositions, or, more generally, the nonnegtive integral coefficients in the decomposition of various symmetric functions in certain bases. The PI aims to determine the behavior of such constants -- asymptotics, positivity, relation to each other, combinatorial interpretation. Among the flagship problems and ultimate goals are: combinatorial interpretation for the Kronecker and plethysm coefficients, Foulkes' conjecture on the relative order of plethysm coefficients, asymptotics of Littlewood-Richardson and Kronecker coefficients, the asymptotic number of skew (semi)Standard Young tableaux in various growth regimes for the parameters, limit behavior of lozenge tilings of "skew" (general, nontrapezoidal) domains, inequalities of multiplicities in Geometric complexity Theory leading to obstructions distinguishing between polynomials, positivity of e-expansions for q-analogues of chromatic symmetric functions and Schur positivity for LLT polynomials. The methods range from extension of existing approaches in the PI's work, to further combination with methods from statistical mechanics like the variational principle, enumeration via large deviations, algebro-geometric interpretations, etc. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.