Related papers: Superintegrability of Kontsevich matrix model

Lehmer constructs four classes of matrices constructed from roots of unity for which the characteristic polynomials and the $k$-th powers can be determined explicitly. Here we study a class of matrices which arise naturally in…

We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral…

A countable set of quantum superintegrable systems for arbitrary spin is solved explicitly using tools of supersymmetric quantum mechanics. It is shown that these systems (introduced by Pronko, J. Phys. A: Math. Theor. 40 (2007) ) include…

We prove that the point process of the eigenvalues of real or complex non-Hermitian matrices $X$ with independent, identically distributed entries is hyperuniform: the variance of the number of eigenvalues in a subdomain $\Omega$ of the…

We consider solvable matrix models. 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…

The polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices (i.e. square matrices such that the differences of their Gram matrices and the corresponding identity matrices are of rank k) in…

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…

The concepts of differentiation and integration for matrices were introduced for studying zeros and critical points of complex polynomials. Any matrix is differentiable, however not all matrices are integrable. The purpose of this paper is…

A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain…

There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity…

A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,\mu<s$ for any other eigenvalue $\mu$. Nonnegative matrices and symmetric ones are typical examples of this…

This paper continues the authors' work on the question of unitary equivalence of matrices with entries in the complex-valued functions of a topological space (matrices over spaces). Specifically, we here consider the question of unitary…

The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I introduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of…

The paper presents the classification of matrix valued superpotentials corresponding to shape invariant systems of Schr\"odinger equations. All inequivalent irreducible matrix superpotentials realized by matrices of arbitrary dimension with…

We analyze the Macdonald's $(q,t)$-deformed hypergeometric functions with one and two set variables and present their constraints. We prove the uniqueness to the solutions of these constraints. We propose a concise method to prove the…

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion…

In this paper we propose a resolution to the problem of $\beta$-deforming the non-Gaussian monomial matrix models. The naive guess of substituting Schur polynomials with Jack polynomials does not work in that case, therefore, we are forced…

We examine and present new combinatorics for the Schur polynomials from the viewpoint of quantum integrability. We introduce and analyze an integrable six-vertex model which can be viewed as a certain degeneration model from a t-deformed…

The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on…

We study the algebraic geometrical background of the Penner--Kontsevich matrix model with the potential $N\alpha \tr {\bigl(- \fr 12 \L X\L X +\log (1-X)+X\bigr)}$. We show that this model describes intersection indices of linear bundles on…