Topological K-Theory for Hilbert Scheme Analogs

TL;DR

This paper computes the topological K-theory of classifying spaces related to Hilbert scheme analogs, bridging a gap between algebraic and topological perspectives in geometric representation theory.

Contribution

It introduces the computation of topological K-theory for classifying spaces associated with Hilbert scheme analogs, extending understanding beyond algebraic methods.

Findings

Computed topological K-theory for classifying spaces like $BS_n$ and $B( ext{G} times S_n)$

Bridged the gap between algebraic and topological approaches in geometric representation theory

Provided new insights into the homotopical aspects of Hilbert scheme analogs.

Abstract

In geometric representation theory, it is common to compute equivariant $K$ theory of schemes like $Hilb^n ( \mathbb{A}^2 )$ or $Hilb^n (X)$ for an ALE resolution $X \to \mathbb{A}^2 / \Gamma$. If we abandon the algebraic nature and just look at this homotopically we see close relatives of $BS_n$ and $B(\Gamma \wr S_n)$. Therefore we compute the topological K theory of these classifying spaces to fill in a small gap in the literature.