Related papers: Macdonald-Hurwitz Number
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…
The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to…
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical…
Monotone Hurwitz numbers were introduced by the authors as a combinatorially natural desymmetrization of the Hurwitz numbers studied in enumerative algebraic geometry. Over the course of several papers, we developed the structural theory of…
Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz…
We prove that single $G$-weighted $\mathfrak{b}$-Hurwitz numbers with internal faces are computed by refined topological recursion on a rational spectral curve, for certain rational weights $G$. Consequently, the $\mathfrak{b}$-Hurwitz…
We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald $P$-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our…
Hurwitz algebras are unital composition algebras widely known in algebra and mathematical physics for their useful applications. In this paper, inspired by works of Lesenby and Hitzer, we show how to embed all seven Hurwitz algebras…
Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized…
In this paper we announce some results obtained for certain algebraic functions, which we call of cyclotomic type. The main results properly resemble von Staudt-Clausen's theorem and Kummer's congruence for the Bernoulli numbers, and such…
We construct $\Delta$-operators $F[\Delta]$ on the space of almost symmetric functions $\mathscr{P}_{as}^{+}$. These operators extend the usual $\Delta$-operators on the space of symmetric functions $\Lambda \subset \mathscr{P}_{as}^{+}$…
Let {\alpha} be a prime Hurwitz integer. H{\alpha}, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of H, which is the set of all Hurwitz integers. We consider left…
We explain how Gaussian integrals over ensemble of complex matrices with source matrices generate Hurwitz numbers of the most general type, namely, Hurwitz numbers with arbitrary orientable or non-orientable base surface and arbitrary…
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a…
We consider the generating series of oriented and non-oriented hypermaps with controlled degrees of vertices, hyperedges and faces. It is well known that these series have natural expansions in terms of Schur and Zonal symmetric functions,…
By applying the derivative operator to the corresponding hypergeometric form of a $q$-series transformation due to Andrews [1,Theorem 4], we establish a general harmonic number identity. As the special cases of it, several interesting…
The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The…
The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The…
The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted…
Multicurrent correlators associated to KP $\tau$-functions of hypergeometric type are used as generating functions for weighted Hurwitz numbers. These are expressed as formal Taylor series and used to compute generic, simple, rational and…