(q,t)-analogues and GL_n(F_q)

TL;DR

This paper introduces a (q,t)-generalization of binomial coefficients that unifies combinatorial and algebraic interpretations, connecting tableaux, Macdonald functions, permutation statistics, and invariant theory of GL_n(F_q).

Contribution

It presents a novel (q,t)-binomial coefficient framework that bridges combinatorics, algebra, and representation theory, extending classical binomial coefficients.

Findings

Defines (q,t)-binomial coefficients with combinatorial and algebraic interpretations.

Establishes connections to Macdonald's seventh variation of Schur functions.

Links permutation statistics and Hilbert series in invariant theory.

Abstract

We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald's ``seventh variation'' of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(F_q).