Shifted genus expanded $\cal{W}_{\infty}$ algebra and shifted Hurwtiz numbers

TL;DR

This paper introduces a new algebraic structure called the shifted genus expanded $\\cal{W}_{\infty}$ algebra, linking it to symmetric group algebra and shifted Schur functions, and applies it to derive differential equations for shifted Hurwitz numbers.

Contribution

It constructs the shifted genus expanded $\cal{W}_{\infty}$ algebra and establishes its isomorphism with known algebraic structures, providing new tools for studying shifted Hurwitz numbers.

Findings

Derived differential equations for generating functions of shifted Hurwitz numbers.

Expressed generating functions using shifted genus expanded cut-and-join operators.

Established isomorphism between the new algebra and existing symmetric function algebras.

Abstract

We construct the shifted genus expanded $\cal{W}_{\infty}$ algebra, which is isomorphic to the central subalgebra $\cal{A}_{\infty}$ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra $\Lambda^\ast$ defined by A. Y. Okounkov and G. I. Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers, thus we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.