Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Limits of group algebras for growing symmetric groups and wreath products

Irina Devyatkova, Grigori Olshanski

TL;DR

This work constructs a robust completion of the group algebra of the infinite symmetric group by introducing a virtual group algebra $\mathcal{B}$, built from limits of finite algebras in the tame representation framework, and proving its isomorphism to a centralizer-based algebra $\mathcal{A}$ arising from Molev–Olshanski’s symmetric inverse semigroups. It extends the approach to wreath products $G\wr S(\infty)$, using a parallel centralizer framework and a generalized semigroup method to obtain $\mathcal{A}(G)$ and $\mathcal{B}(G)$, with an isomorphism $\mathcal{B}(G)\cong\mathcal{A}(G)$. The paper also draws deep parallels with infinite-dimensional classical groups, where the center is captured by shifted symmetric functions $\operatorname{Sym}^*$ and a Yangian-like factor appears, highlighting a unifying A-type duality across discrete and continuous settings. The resulting structure provides a countable presentation, clarifies the dimensionality, and yields a natural, constructive analogue of group algebras for these infinite objects, with potential implications for the representation theory of large, combinatorially structured groups. Overall, the work broadens the toolkit for handling tame representations in infinite settings and connects several algebraic frameworks via a unifying centralizer- and semigroup-based perspective.

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 $S(\infty)$. We establish a connection with the centralizer construction of Molev-Olshanski [J. Algebra, 237 (2001), 302-341; arXiv:math/0002165] and Drinfeld-Lusztig degenerate affine Hecke algebras. This makes it possible to describe the structure of the virtual group algebra. Then we extend the results to wreath products $G\wr S(\infty)$ with arbitrary finite groups $G$.

Limits of group algebras for growing symmetric groups and wreath products

TL;DR

This work constructs a robust completion of the group algebra of the infinite symmetric group by introducing a virtual group algebra , built from limits of finite algebras in the tame representation framework, and proving its isomorphism to a centralizer-based algebra arising from Molev–Olshanski’s symmetric inverse semigroups. It extends the approach to wreath products , using a parallel centralizer framework and a generalized semigroup method to obtain and , with an isomorphism . The paper also draws deep parallels with infinite-dimensional classical groups, where the center is captured by shifted symmetric functions and a Yangian-like factor appears, highlighting a unifying A-type duality across discrete and continuous settings. The resulting structure provides a countable presentation, clarifies the dimensionality, and yields a natural, constructive analogue of group algebras for these infinite objects, with potential implications for the representation theory of large, combinatorially structured groups. Overall, the work broadens the toolkit for handling tame representations in infinite settings and connects several algebraic frameworks via a unifying centralizer- and semigroup-based perspective.

Abstract

Let 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 . The virtual group algebra is obtained by taking large- limits of the finite-dimensional group algebras in the so-called tame representations of . We establish a connection with the centralizer construction of Molev-Olshanski [J. Algebra, 237 (2001), 302-341; arXiv:math/0002165] and Drinfeld-Lusztig degenerate affine Hecke algebras. This makes it possible to describe the structure of the virtual group algebra. Then we extend the results to wreath products with arbitrary finite groups .

Paper Structure

This paper contains 63 sections, 45 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Statement of problem
  3. The virtual group algebra of $S(\infty)$
  4. The centralizer construction
  5. Main result (case of $S(\infty)$)
  6. Generalization to wreath products
  7. Analogy with infinite-dimensional classical groups
  8. Organization of the paper
  9. Tame representations of $S(\infty)$
  10. Definition of tame representations
  11. The irreducible representations $T^\lambda$
  12. The spectrum of $T^\lambda|_{S(n)}$
  13. Another realization of representations $T^\lambda$
  14. Analog of Young's branching rule for $T^\lambda$
  15. Virtual center of the group algebra $\mathbb C[S(\infty)]$
  16. ...and 48 more sections

Key Result

Theorem 1.1

The algebra ${\mathcal{B}}$ (the virtual group algebra of $S(\infty)$) is isomorphic to the algebra ${\mathcal{A}}$ defined by the centralizer construction.

Theorems & Definitions (103)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5: Ols80, Theorem 2.7; MO01, Proposition 2.5
  • proof
  • Proposition 2.6: Branching rule
  • ...and 93 more