Algebras of conjugacy classes in symmetric groups and checker triangulated surfaces

Yu. A. Neretin<sup>abcd

a</sup> Math. Dept., University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien
^b Institute for Information Transmission Problems
^c Institute for Theoretical and Experimental Physics (until 11.2021)
^d Mech. Math. Dept., Moscow State University

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, $G=S_n \times S_n$, $K$ is the diagonal subgroup $S_n$. In this case we get a geometric description of this algebra.

Key words and phrases: symmetric groups, group algebras, Ivanov–Kerov algebra, partial bijections, triangulated surfaces, conjugacy classes.

MSC: 20B30, 20C32, 20E45

Language: English

DOI: 10.17323/1609-4514-2024-24-2-201-217