Related papers: Determinantal Representations and B\'ezoutians

The continuous big $q$-Hermite polynomials are shown to realize a basis for a representation space of an extended $q$-oscillator algebra. An expansion formula is algebraically derived using this model.

Can a smooth plane cubic be defined by the determinant of a square matrix with entries in linear forms in three variables? If we can, we say that it admits a linear determinantal representation. In this paper, we investigate linear…

We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our…

The quantum $\alpha$-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic $\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra $\mathcal{A}_q(\mathrm{Mat}_2)$…

There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also…

This paper determines the representation type of the Iwahori-Hecke algebras of type B when $q\ne\pm1$. In particular, we show that a single parameter non-semisimple Iwahori-Hecke algebra of type B has finite representation type if and only…

Two types of determinant representations of the rational solutions for the Painlev\'e II equation are discussed by using the bilinear formalism. One of them is a representation by the Devisme polynomials, and another one is a Hankel…

For a smooth hypersurface of even dimension, the quadratic character of the absolute Galois group defined by the determinant of the l-adic cohomology of middle dimension is computed via the square root of the discriminant of a defining…

We give simple proofs for the Hankel determinants of q-exponential polynomials.

A hypersurface $H$ in $\mathbb{P}^r$ of degree $n$ is called determinantal if it is the zero locus of a polynomial of the form $\operatorname{det}(x_0 A_0 + \ldots + x_r A_r)$ for some $(r+1)$-tuple of $n \times n$ matrices $A = (A_0,…

We describe a bilinear identity satisfied by certain multidimensional q-hypergeometric integrals. The identity can be considered as a deformation of the Riemann bilinear relation for the twisted de Rham (co)homologies. The identity also…

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic…

We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of…

The derivative polynomials for the hyperbolic and trigonometric tangent, cotangent and secant are found in explicit form, where the coefficients are given in terms of Stirling numbers of the second kind. As application, some integrals are…

We show that the moduli space M of marked cubic surfaces is biholomorphic to the quotient by a discrete group generated by complex reflections of the complex four-ball minus the reflection hyperplanes of the group. Thus M carries a complex…

There are many similarities between the theories of matroids and $q$-matroids. However, when dealing with the direct sum of $q$-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable…

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form…

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod…

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell…

We give a bound on the number of isolated, essential singularities of determinantal quartic surfaces in 3-space. We also provide examples of different configurations of real singularities on quartic surfaces with a definite Hermitian…