Connection between cut-and-join and Casimir operators

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.

14 pages

References in corpus (7)

• Generation of Matrix Models by W-operators
• Matrix Models for Random Partitions
• Hurwitz numbers, matrix models and enumerative geometry
• The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations
• Hypergeometric functions related to Schur Q-polynomials and BKP equation
• Recursion Formulas for Spin Hurwitz Numbers
• Notes about KP/BKP correspondence

Cited by in corpus (7)

• Spin Hurwitz theory and Miwa transform for the Schur Q-functions
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Generalized $\tilde{W}$ algebras
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• q-Casimir and q-cut-and-join operators related to Reflection Equation Algebras
• Notes about KP/BKP correspondence
• A new kind of anomaly: on W-constraints for GKM