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Algebras of conjugacy classes in symmetric groups

Yury A. Neretin

Abstract

In 1999 V. Ivanov and S. Kerov observed that structure constants of algebras of conjugacy classes of symmetric groups $S_n$ admit a stabilization (in a non-obvious sense) as $n\to \infty$. We extend their construction to a class of pairs of groups $G\supset K$ and algebras of conjugacy classes of $G$ with respect to $K$. In our basic example $G$ is a product of symmetric groups, $G=S_n \times S_n$, $K$ is the diagonal subgroup $S_n$.

Algebras of conjugacy classes in symmetric groups

Abstract

In 1999 V. Ivanov and S. Kerov observed that structure constants of algebras of conjugacy classes of symmetric groups admit a stabilization (in a non-obvious sense) as . We extend their construction to a class of pairs of groups and algebras of conjugacy classes of with respect to . In our basic example is a product of symmetric groups, , is the diagonal subgroup .

Paper Structure

This paper contains 3 sections, 5 theorems, 78 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A more general construction
  3. Final remarks

Key Result

Theorem \oldthetheorem

Let ${\,{\overline{\overline {g\newline}}}\,}$, ${\,{\overline{\overline {h\newline}}}\,}$, ${\,{\overline{\overline {r\newline}}}\,}$ range in the disjoint union $\coprod_{j=0}^\infty G_j/\!\!/K_j$. Then there are non-negative integers $a_{{\,{\overline{\overline {g\newline}}}\,},{\,{\overline{\ove $\bullet$ Consider a linear space $\mathcal{B}$ with a basis consisting of symbols $A[{\,{\overline

Figures (6)

  • Figure 1: a) A piece of a checker triangulated surface. b) A piece of a labeled checker triangulated surface.
  • Figure 2: A surface consisting of two triangles.
  • Figure 3: Surfaces $\mathcal{R}$, $\mathcal{Q}$ and a partial bijection $\mathcal{R}_-$ to $\mathcal{Q}_+$.
  • Figure 4: A surface whose vertices are glued.
  • Figure 5: Sets $V\supset V_n$.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem