Connection between cut-and-join and Casimir operators

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. (C) 2021 The Author(s). Published by Elsevier B.V.

Automated summary

We study cut-and-join operators for spin Hurwitz partition functions and provide explicit expressions for these operators in terms of derivatives in p-variables. This allows for direct and algorithmic calculation, facilitated by the connection between the operators and easy-to-compute Casimir operators. An essential part of this connection involves shifted Q-Schur functions.