Research Papers

Limits of group algebras for growing symmetric groups and wreath products

I. E. Devyatkova^ab, G. I. Olshanski<sup>ab

a</sup> Skolkovo Institute of Science and Technology
^b National Research University Higher School of Economics

Abstract: Let $S(\infty)$ denote the infinite symmetric group formed by the finitary permutations of the set of natural numbers; this is a countable group. We introduce its virtual group algebra, a completion of the conventional group algebra $\mathbb C[S(\infty)]$. The virtual group algebra is obtained by taking large-$n$ limits of the finite-dimensional group algebras $\mathbb C[S(n)]$ in the so-called tame representations of $S(\infty)$. (Note that our virtual group algebra is very different from the $C^*$-envelope.) We describe the structure of the virtual group algebra, which reveals a connection with Drinfeld-Lusztig degenerate affine Hecke algebras. Then we extend the results to wreath products $G\wr S(\infty)$ with arbitrary finite groups $G$.

Keywords: infinite symmetric group, tame representations, degenerate affine Hecke algebra.

Received: 09.04.2025