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 with Rin hbox {SP} are common eigenfunctions of cut-and-join operators W_Delta with Delta in hbox {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.
Highlights
Note that the original construction essentially involves the characters of linear groups and symmetric groups understood as embedded into the linear group G L(∞) and the symmetric group S∞
Instead of the Schur polynomials we deal with the Q Schur functions, and instead of the symmetric groups we deal with the Sergeev groups
A peculiar property of symmetric polynomials from Macdonald family is that the sum at the r.h.s is restricted from the naive R1 + R2 ≤ R ≤ R1 ∪ R2 in the lexicographical ordering to a narrower sum of irreducible representations of S L N emerging in the tensor product of representations associated with the Young diagrams R1 and R2: R ∈ R1 ⊗ R2
Summary
Note that the original construction essentially involves the characters of linear groups and symmetric groups (another manifestation of the Schur–Weyl duality) understood as embedded into the linear group G L(∞) and the symmetric group S∞. We discuss this interesting subject with the hope that it would add essential new colors to the picture and give rise to many new applications. Instead of the Schur polynomials (characters of linear groups) we deal with the Q Schur functions, and instead of the symmetric groups we deal with the Sergeev groups. Immediate subjects to address within this context are more or less standard, we list them in the table below indicating where they are discussed in this paper: Subject. W to (shifted) symmetric functions Integrability Matrix models and character expansions
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