Related papers: Powers of the Vandermonde determinant, Schur Funct…
In this note, we apply the power-partible reduction to show the following arithmetic properties of large Schr\"oder polynomials $S_n(z)$ and little Schr\"oder polynomials $s_n(z)$: for any odd prime $p$, nonnegative integer…
we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable…
The notion of a descent polynomial, a function in enumerative combinatorics that counts permutations with specific properties, enjoys a revived recent research interest due to its connection with other important notions in combinatorics,…
We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter $\nu>0$. The LDU-decomposition of the weight is explicitly given in…
The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from…
The Springer modules have a combinatorial property called ``coincidence of dimensions,'' i.e., the Springer modules are naturally decomposed into submodules with common dimensions. Morita and Nakajima proved the property by giving modules…
We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…
We show that certain determinantal functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. These are shown to be true in complete generality; that is, no properties of…
The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed…
In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit…
We compute the generating function for the characters of the irreducible representations of SU(n) whose associated Young diagrams have only two rows with the same number of boxes. The result is a rational determinantal expression in which…
We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew…
We wonder if there is a way to make all Schur functions in all representations equal. This is impossible for fixed value of time variables, but can be achieved for averages. It appears that the corresponding measure is just Gaussian in…
We give geometric descriptions of the category C_k(n,d) of rational polynomial representations of GL_n over a field k of degree d for d less than or equal to n, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a…
Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with…
Introduced by Okounkov and Reshetikhin, the Schur process is known to be a determinantal point process, meaning that its correlation functions are minors of a single correlation kernel matrix. Previously, this was derived using…
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur…
We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric…
We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot \mathbf{1}^c)/(\mathbf{\lambda} + k\cdot…
Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the…