Related papers: From $r$-Spin Intersection Numbers to Hodge Integr…

We reexamine the external field problem for $N\times N$ hermitian one-matrix models. We prove an equivalence of the models with the potentials $\tr{({1/over2N}X^2 + \log X - \Lambda X)}$ and $\sum_{k=1}^\infty t_k\tr{X^k}$ providing the…

We introduce a class of generalized Lipkin-Meshkov-Glick (gLMG) models with su$(m)$ interactions of Haldane-Shastry type. We have computed the partition function of these models in closed form by exactly evaluating the partition function of…

A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear…

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials considered by S.Milne \cite{Milne}. For $q=1$ these functions are also known as hypergeometric functions of matrix argument…

We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the…

The principal goal of the paper is to apply the approach inspired by the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on…

We revisit a double-scaled limit of the superconformal index of ${\cal N}=2$ superconformal field theories (SCFTs) which generalizes the Schur index. The resulting partition function, $\hat {\cal Z}(q,\alpha)$, has a standard $q$-expansion…

The permutation matrices form a subgroup of $\text{GL}_n(\mathbb{C})$ that is isomorphic to the symmetric group $S_n$. Let $r_{\mu\lambda}$ denote the multiplicity of the irreducible representation $V_\mu$ of $S_n$, corresponding to a…

We prove combinatorially the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for…

We show that partition functions of various matrix models can be obtained by acting on elementary functions with exponents of W-operators. A number of illustrations is given, including the Gaussian Hermitian matrix model, Hermitian model in…

One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{\lambda \, \mu \, \nu}$. A notable advance in this direction is due to…

The determinantal form of the partition function of the 6-vertex model with domain wall boundary conditions was given by Izergin. It is known that for a special value of the crossing parameter the partition function reduces to a Schur…

The quantum spectral curve equation associated to KP $\tau$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning…

We study matrix integration over the classical Lie groups $U(N),Sp(2N),O(2N)$ and $O(2N+1)$, using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz$\pm$Hankel matrices. We establish a…

An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{\xi\mu}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more…

The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov-Schulman operators turn out to satisfy a set of generalized string equations. These generalized string…

We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In…

Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized…

We use the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM), which is the generating function for volumes of discretized…

We continue the construction of a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with an arbitrary Young tableaux having $k$ rows, on a basis of the BRST--BFV approach suggested for…