Related papers: Character expansion of matrix integrals
We present an explicit formula for subregular characters (i.e, irreducible finite-dimensional complex characters of submaximal degree) of the unitriangular group over a finite field of sufficiently large characteristic.
These notes are our contribution to the Proceedings of the ICM 2026. We discuss some results we have obtained (in part jointly with coauthors) regarding the representation theory of reductive algebraic groups over algebraically closed…
We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur…
A new family of strongly regular graphs, called the general symplectic graphs $Sp(2\nu, q)$, associated with nonsingular alternate matrices is introduced. Their parameters as strongly regular graphs, their chromatic numbers as well as their…
We consider integrals of tau functions of Zakharov-Shabat systems whose higher times are related to the eigenvalues of products of random matrices. Apart of random matrices there is the set of $n$ pairs of given matrices which play the role…
The paper explores various special functions which generalize the two-parametric Mittag-Leffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values…
For a graph consisting of parallel connected subgraphs we express the characteristic function of the boundary value problem with generalized Neumann conditions at both joining points via characteristic functions of different boundary…
Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are…
We construct some canonically defined central extensions of groups of symplectomorphisms. We show that this central extension is nontrivial in the case of a torus of dimension $\ge 6$ and in the case of a two-dimensional surface of genus…
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…
A general setting to study a certain type of formulas, expressing characters of the symmetric group $\mathfrak{S}_n$ explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is…
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…
We obtain two explicit formulas for the full local character expansion of any irreducible representation of a p-adic general linear group in principal blocks. The first, generalizing previous work of the author on the Iwahori-spherical…
We study integrals over Hermitian supermatrices of arbitrary size $p+q$, that are parametrized by an external field $X$ and a source $Y$, of respective size $m+n$ and $p+q$. We show that these integrals exhibit a simple topological…
Turull has described the fields of values for characters of $SL_n(q)$ in terms of the parametrization of the characters of $GL_n(q)$. In this article, we extend these results to the case of $SU_n(q)$.
In this paper we give a complete description of the Howe correspondence of unipotent characters for a finite dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization under a mild restriction of…
We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical $r$-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the…
The area related to M. Liv\v{s}ic's characteristic matrix functions is too vast to be discussed in one paper and we selected for this article the problems which are close to our scientific interests. We discuss M.Liv\v{s}ic's results…
We introduce two vertex operators to realize skew odd orthogonal characters $so_{\lambda/\mu}(x^{\pm})$ and derive the Cauchy identity for the skew characters via Toeplitz-Hankel-type determinant similar to the Schur functions. The method…
Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $\Delta(X) = X\otimes X$, then their generating functions obey…