Related papers: Connection between cut-and-join and Casimir operat…
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…
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…
We construct a cubic cut-and-join operator description for the partition function of the Chekhov-Eynard-Orantin topological recursion for a local spectral curve with simple ramification points. In particular, this class contains partition…
Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary $N\times N$…
In this short note we construct a simple cut-and-join operator representation for Kontsevich-Witten tau-function that is the partition function of the two-dimensional topological gravity. Our derivation is based on the Virasoro constraints.…
To distinguish the contributions to the generalized Hurwitz number of the source Riemann surface with different genus, we define the genus expanded cut-and-join operators by observing carefully the symplectic surgery and the gluing formulas…
Goulden and Jackson first introduced the cut-and-join operator. The cut-and-join is widely used in studying the Hurwitz number and many other topological recursion problems. Mironov, Morosov and Natanzon give a more general construction and…
We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. They are related to peculiar $Q$ Schur functions, which are actually related to characters of the Sergeev group. This allows one to…
We show the integrality of the simple Hurwitz numbers. The main tool is the cut-and-join operator, and our proof is a purely combinatorial one.
We extend the old formalism of cut-and-join operators in the theory of Hurwitz $\tau$-functions to description of a wide family of KP-integrable {\it skew} Hurwitz $\tau$-functions, which include, in particular, the newly discovered…
We demonstrate in some detail how Macdonald polynomials emerge from the recently introduced 3-Schur functions when the plane-partition vector time-variables are projected onto the ordinary scalar times under non-vanishing angles, which…
We construct a family of bilinear differential operators which satisfy certain gauge properties. These operators can be naturally associated with $q$-deformations of classical integrable hierarchies. In particular, we consider the case when…
We follow the general recipe for constructing commutative families of $W$-operators, which provides Hurwitz-like expansions in symmetric functions (Macdonald polynomials), in order to obtain a difference operator example that gives rise to…
Young diagrams can be parameterized with the help of hook variables, which is well known but never studied in big detail. We demonstrate that this is the most adequate parameterization for many physical applications: from the Schur…
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…
Schur's $Q$-functions with reduced variables are discussed by employing a combinatorics of strict partitions. They are called reduced $Q$-functions. We give a description of the linear relations among reduced $Q$-functions.
On a finite graph, we prove that trace of holonomies determine an intertwining relation between merge-and-split generators on collections of geodesic loops ensembles and Casimir operators on unitary connections. By adding a deformation part…
The quadratic Casimir operator of the special unitary $SU(N)$ group is used to construct projection operators, which can decompose any of its reducible finite-dimensional representation spaces contained in the tensor product of two and…
W-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the W-operator to prove a formula for the generating function of the generalized Hurwitz…
A new combinatorial approach to the ribbon tableaux generating functions and q-Littlewood Richardson coefficients of Lascoux, Leclerc and Thibon is suggested. We define operators which add ribbons to partitions and following Fomin and…