Asymptotic behaviors in the homology of symmetric group and finite general linear group quandles

TL;DR

This paper investigates the homology of quandles derived from symmetric and finite general linear groups using representation theory, introducing FI- and VIC$(q)$-quandle structures and constructing related link invariants.

Contribution

It introduces FI- and VIC$(q)$-quandle structures on conjugacy classes and applies representation theory to analyze their homology and link invariants.

Findings

Homology of symmetric group and finite general linear group quandles analyzed.

Construction of FI-module and VIC$(q)$-module invariants of links.

Proved that these conjugacy class collections form FI- and VIC$(q)$-quandles.

Abstract

A quandle is an algebraic structure which attempts to generalize group conjugation. These structures have been studied extensively due to their connections with knot theory, algebraic combinatorics, and other fields. In this work, we approach the study of quandles from the perspective of the representation theory of categories. Namely, we look at collections of conjugacy classes of the symmetric groups and the finite general linear groups, and prove that they carry the structure of FI-quandles (resp. VIC$(q)$-quandles). As applications, we prove statements about the homology of these quandles, and construct FI-module and VIC$(q)$-module invariants of links.