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

Abstract

We construct the shifted genus expanded ${\mathcal{W}}_{\infty }$ algebra, which is isomorphic to the central subalgebra ${\mathcal{A}}_{\infty }$ 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.

16 pages, no figure, any comments welcome! arXiv admin note: text overlap with arXiv:1511.00823

References in corpus (5)

• Generation of Matrix Models by W-operators
• On KP-integrable Hurwitz functions
• Algebra of differential operators associated with Young diagrams
• The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations
• Genus Expanded Cut-and-Join operators and generalized Hurwtiz numbers