Cut-and-join structure and integrability for spin Hurwitz numbers
A. Mironov, A. Morozov, S. Natanzon
Abstract
Abstract Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions $$Q_R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi><mml:mi>R</mml:mi></mml:msub></mml:math> with $$R\in \hbox {SP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>∈</mml:mo><mml:mtext>SP</mml:mtext></mml:mrow></mml:math> are common eigenfunctions of cut-and-join operators $$W_\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>W</mml:mi><mml:mi>Δ</mml:mi></mml:msub></mml:math> with $$\Delta \in \hbox {OP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Δ</mml:mi><mml:mo>∈</mml:mo><mml:mtext>OP</mml:mtext></mml:mrow></mml:math> . The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>τ</mml:mi></mml:math> -function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.