Related papers: Hurwitz numbers, ribbon graphs, and tropicalizatio…
Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The quantum double of each such Hopf algebra is investigated. The quantum doubles of a family of Hopf algebras from the above…
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic…
We prove a tropical analogue of the theorem of Hurwitz: a leafless metric graph of genus $g \ge 2$ has at most $12$ automorphisms when $g = 2$; $2^g g!$ automorphisms when $g \ge 3$. These inequalities are optimal; for each genus, we give…
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…
This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and…
In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating…
We give combinatorial proofs of some enumeration formulas involving labelled threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs. In each case we count by number of vertices and number of components. For threshold…
We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type. These operators have two alternative descriptions:(i) they have the GL characters as eigenfunctions and the symmetric-group characters…
Recently a new family of enumerative invariants called leaky Hurwitz numbers was introduced by Cavalieri-Markwig-Ranganathan in the context of logarithmic intersection theory. They admit an interpretation via tropical covers where the…
Multizeta values are real numbers which span a complicated algebra: there exist two different interacting products. A functional analog of these numbers is defined so as to obtain a better understanding of them, the Hurwitz multizeta…
Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs…
The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ of a ribbon graph $G$ is the generating function that enumerates all partial duals of $G$. In this paper, we give a categorification for this polynomial. The key ingredient of…
The theory of representations of a crossed module is a direct generalization of the theory of representations of groups. For a finite group G, the Drinfeld quantum double of the group G is a Hopf algebra that represents a special case of…
We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define "combinatorial Hodge integrals" from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
We derive the spectral curves for $q$-part double Hurwitz numbers, $r$-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0,1)-geometry. We quantize this family of spectral curves and…
We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of…
Biserial algebras are a classical class in the representation theory of algebras, generalizing Nakayama algebras. They were further generalized by Green and Schroll to multiserial algebras, which share many structural properties with…
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We…
Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincar\'e duality. The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ is an enumeration of the…