A Combinatorial Proof of a Recursion for the q-Kostka Polynomials

The Kostka numbers Kλμ are important in several areas of mathematics, including symmetric function theory, representation theory, combinatorics and invariant theory. The q-Kostka polynomials Kλμ(q) are the q-analogues of the Kostka numbers. They generalize and extend the mathematical meaning of the Kostka numbers. Lascoux and Schützenberger proved one can attach a non-negative integer statistic called charge to a semistandard tableau of shape λ and content μ such that Kλμ(q) is the generating function for charge on those semistandard tableaux. We will give two new descriptions of charge and prove several new properties of this statistic. In addition, the q-Kostka polynomials are known to satisfy a certain shape and content reducing recursion. We will give a combinatorial proof of a related recursion for the q-Kostka polynomials on words.