k-partial permutations and the center of the wreath product $${\mathcal {S}}_k\wr {\mathcal {S}}_n$$ algebra

Abstract

AbstractWe 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 $${\mathcal {S}}_k\wr {\mathcal {S}}_n$$ S k ≀ S n algebra are polynomials in n with nonnegative integer coefficients. We use a universal algebra $${\mathcal {I}}_\infty ^k$$ I ∞ k , which projects on the center $$Z({\mathbb {C}}[{\mathcal {S}}_k\wr {\mathcal {S}}_n])$$ Z ( C [ S k ≀ S n ] ) for each n. We show that $${\mathcal {I}}_\infty ^k$$ I ∞ k is isomorphic to the algebra of shifted symmetric functions on many alphabets.