Algebraic, Combinatorial, and Analytic Applications of Symmetric Functions

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From the lattice structure of crystals, to states of matter, to matrices and differential operators, the traits of systems and their evolution are classified by symmetries. Algebraic combinatorics studies symmetries via their manifestations in well-known discrete objects like graphs, permutations, and partitions. Its methods have successfully solved problems in other sciences such as physics, computer sciences, and biology. This project concerns the application of algebraic combinatorics, in particular its subfield the theory of symmetric functions, to solve such problems. This project is centered around the tools used, namely, the theory of symmetric functions and the associated combinatorics. Various complexity problems in representation theory concern the computation of certain structure constants and multiplicities that are expressible via the Kronecker and plethystic coefficients of the symmetric group, which can be defined using Schur functions. In statistical mechanics, the partition functions of some integrable lattice models like lozenge tilings are often Lie group characters, and their asymptotic study reveals probabilistic behavior like Gaussian unitary ensemble eigenvalue distribution near the boundary or the existence of limit shapes and surfaces. This project aims to expand these applications to study other models and distributions. Studying combinatorial and algebraic properties of Schubert polynomials, as representatives of the cohomology classes of flag varieties, can lead to combinatorial interpretations for the corresponding structure constants. Further, computational properties of their stable versions, the Stanley symmetric functions, could lead to understanding of the mysterious limit behavior of random sorting networks, corresponding to the reduced decompositions of permutations into adjacent transpositions.