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

The Kostka numbers Kλμ play an important role in symmetric function theory, representation theory, combinatorics and invariant theory. The q-Kostka polynomials Kλμ(q) are the q-analogues of the Kostka numbers and 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 the Kostka polynomial 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. These new descriptions of charge will be used to give a combinatorial proof of a content reducing recursion for the q-Kostka polynomials originally proved by A. M. Garsia and C. Procesi (1992, Adv. in Math.94, 82–138).