On permutation statistics, root enumeration and Gelfand models

The k-th root enumerator of a finite group G is an integer valued function on G, which counts the number of k-th roots of each element. A long-standing open problem is to classify the finite groups, for which the k-th root enumerator is a proper (non-virtual) character for all k. Another well known problem is to construct Gelfand models; these are multiplicity-free sums of all the irreducible characters of a finite group.

It will be shown that for all classical Weyl groups, all k-th root enumerators are proper, extending the results of Scharf and Thibon. The proof is constructive and presents the root enumerator as a multiplicity-free sum of higher Lie characters. Related constructions of Gelfand models for classical and affine Weyl groups will be presented. Applications to permutation statistics will be described.

Based on joint works with Ron Adin and Pal Hegedus.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/