K-partial permutations and the center of the wreath product {mathcal {S}}_kwr {mathcal {S}}_n algebra

Abstract

We generalize the concept of partial permutations of Ivanov and Kerov and introduce k-partial permutations. This allows us to show that the structure coefficients of the center of the wreath product {mathcal {S}}_kwr {mathcal {S}}_n algebra are polynomials in n with nonnegative integer coefficients. We use a universal algebra {mathcal {I}}_infty ^k, which projects on the center Z({mathbb {C}}[{mathcal {S}}_kwr {mathcal {S}}_n]) for each n. We show that {mathcal {I}}_infty ^k is isomorphic to the algebra of shifted symmetric functions on many alphabets.