Related papers: A Linear System of Differential Equations Related …
A matrix-valued measure $\Theta$ reduces to measures of smaller size if there exists a constant invertible matrix $M$ such that $M\Theta M^*$ is block diagonal. Equivalently, the real vector space ${\mathscr A}$ of all matrices $T$ such…
In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials…
In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the…
In this paper we continue studying of matrix $n\times n$ linear differential intertwining operators. The problems of minimization and of reducibility of matrix intertwining operators are considered and criterions of weak minimizability and…
By using the Hadamard matrix product concept, this paper introduces two generalized matrix formulation forms of numerical analogue of nonlinear differential operators. The SJT matrix-vector product approach is found to be a simple,…
An equivariant linear system on a toric variety is a linear system invariant under the torus action. We study the number of irreducible components of the complete intersection of general divisors from a fixed collection of equivariant…
In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The…
Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by…
Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the…
Jack polynomials in superspace, orthogonal with respect to a ``combinatorial'' scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an ``analytical'' scalar product,…
The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in $n+m$ variables, which reduce to the Jack polynomials when $n=0$ or $m=0$ and provide joint eigenfunctions of the quantum integrals of the…
A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin-Kasteleyn random…
Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this…
The rational Cherednik algebra $\HH$ is a certain algebra of differential-reflection operators attached to a complex reflection group $W$. Each irreducible representation $S^\lambda$ of $W$ corresponds to a standard module $M(\lambda)$ for…
The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case…
We consider positive Jacobi matrices $J$ with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their…
A recent novel derivation of the representation of Virasoro singular vectors in terms of Jack polynomials is extended to the supersymmetric case. The resulting expression of a generic super-Virasoro singular vector is given in terms of a…
For certain negative rational numbers k0, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter k = k0. It was shown by de…
This work investigates a class of non-autonomous $T$-periodic piecewise smooth differential systems and their associated time-$T$ maps. Our main result provides an analytical approach for detecting, within this class of piecewise…
This note presents some equalities in law for $Z_N:=\det(\Id-G)$, where $G$ is an element of a subgroup of the set of unitary matrices of size $N$, endowed with its unique probability Haar measure. Indeed, under some general conditions,…