Deformed universal characters for classical and affine algebras

TL;DR

This paper introduces deformed creation operators for universal characters of classical and affine algebras, producing symmetric functions with positive polynomial expansions and conjectured connections to crystal bases and Macdonald duality.

Contribution

It develops a unified framework for classical and affine types using deformed operators, linking symmetric functions to crystal bases and polynomial positivity.

Findings

Operators produce symmetric functions with non-negative polynomial coefficients.

Conjecture links these polynomials to crystal bases of affine algebra modules.

Polynomials satisfy a Macdonald-type duality.

Abstract

Creation operators are given for the three distinguished bases of the type BCD universal character ring of Koike and Terada yielding an elegant way of treating computations for all three types in a unified manner. Deformed versions of these operators create symmetric function bases whose expansion in the universal character basis, has polynomial coefficients in $q$ with non-negative integer coefficients. We conjecture that these polynomials are one-dimensional sums associated with crystal bases of finite-dimensional modules over quantized affine algebras for all nonexceptional affine types. These polynomials satisfy a Macdonald-type duality.