Orthogonal polynomials

Combinatorics

Physics

Eigenfunction

Pure mathematics

Classical orthogonal polynomials

Mathematics

Evolutionary biology

Gegenbauer polynomials

Biology

Matrix (chemical analysis)

Algebra over a field

Schur's lemma

Symmetric group

Rank (graph theory)

Function (biology)

Eigenvalues and eigenvectors

Composite material

Hurwitz matrix

Materials science

Generating function

Join (topology)

Group (periodic table)

Quantum mechanics

The European Physical Journal C

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 with R\in SP are common eigenfunctions of cut-and-join operators W_\Delta with \Delta\in OP. 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-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

Cut-and-join structure and integrability for spin Hurwitz numbers