Connection between cut-and-join and Casimir operators

TL;DR

This paper explores the relationship between cut-and-join operators for spin Hurwitz functions and Casimir operators, providing explicit derivative expressions and an algorithmic approach based on their connection.

Contribution

It introduces explicit derivative-based expressions for spin cut-and-join operators and links them to Casimir operators using shifted Q-Schur functions, enabling direct computation.

Findings

Explicit derivative expressions for spin cut-and-join operators

Connection established between these operators and Casimir operators

Algorithmic calculation method developed

Abstract

We study cut-and-join operators for spin Hurwitz partition functions. We provide explicit expressions for these operators in terms of derivatives in $p$-variables without straightforward matrix realization, which is yet to be found. With the help of these expressions spin cut-and-join operators can be calculated directly and algorithmically. The reason why it is possible is the connection between mentioned operators and specially chosen Casimir operators that are easy to compute. An essential part of the connection involves shifted Q-Schur functions.