"On the Kronecker Powers of s_{d,d}"

Guoce Xin
University of Kentucky

Wednesday, April 26, 2006
4:00 pm, 845 Patterson Office Tower

Abstract:

For two given integers k,d \geq 1, let c_d(k)= \sum_\rho \frac{\left(\chi_\rho^{(d,d)}\right)^k}{z_\rho}, where \chi_\rho^{(d,d) denotes the value of Young's irreducible character indexed by (d,d) at the permutations of S_{2d} with cycle structure \rho, and n!/z_\rho gives the size of the corresponding conjugacy class.

The Wallach series $W_k(q) = 1+ \sum_{d\ge 1} c_d(k) q^d$ was created by Nolan Wallach. It has a nice expression for k = 1, 2, 3, 4 and a large expression for k = 5. The k < 5 cases were done by using invariant theory, symmetric function theory, and by a purely combinatorial approach. The k = 5 case was first predicted by Wallach. I will show several ways to compute the Wallach series of the k = 5 case. Our starting point is a related problem of counting solutions of a linear Diophantine system: assign a nonnegative integer label for each vertex of a k-cube such that the sum of the labels for every face are equal. This is a joint work with A. Garsia, G. Musiker, N. Wallach, and M. Zabrocki.