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

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 algebra, which is isomorphic to the central subalgebra of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra 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.
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