| The Kostka numbers Klambdamu are important in several areas of mathematics, including symmetric function theory, representation theory, combinatorics and invariant theory. The q-Kostka polynomials Klambdamu(q) are the q-analogues of the Kostka numbers. They generalize and extend the mathematical meaning of the Kostka numbers. Lascoux and Schutzenberger proved one can attach a non-negative integer statistic called charge to a semistandard tableau of shape lambda and content mu such that Klambdamu(q) is the generating function for charge on those semistandard tableaux. In this thesis, I 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. I will give a combinatorial proof of a related recursion for the q-Kostka polynomials on words. |
