Related papers: Obstructions to determinantal representability

We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…

We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace…

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set…

This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex…

Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal…

We investigate representations of K\"ahler groups $\Gamma = \pi_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a…

Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{\'{e}}'s result on the dimension of vector spaces of square matrices annihilating the…

In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated…

We prove that if M is a vertically 4-connected matroid with a modular flat X of rank at least three, then every representation of M | X over a finite field F extends to a unique F-representation of M. A corollary is that when F has order q,…

With this work we initiate a study of the representations of a unipotent group over a field of characteristic zero from the modular point of view. Let $G$ be such a group. The stack of all representations of a fixed finite dimension $n$ is…

Associated varieties are geometric objects appearing in infinite-dimensional representations of semisimple Lie algebras (groups). By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie…

We prove that supports of non-symmetric Macdonald polynomials are $M$-convex. As a consequence, we resolve a 2019 conjecture of Monical, Tokcan, and Yong that they have the saturated Newton polytope property. As a corollary we show that…

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a…

For an odd prime $p$, we realize the trivial representation of $\mathrm{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ on the free $\mathbb{Z}/p^n \mathbb{Z}$-module of rank one as a subquotient of a direct sum of symmetric power representations (twisted…

Given a combinatorial optimization problem $\Pi$ and an increasing finite sequence $c$ of natural numbers, we obtain a cardinality constrained version $\Pi_c$ of $\Pi$ by permitting only those feasible solutions of $\Pi$ whose cardinalities…

The problem of determining the representation type of the full transformation monoid was resolved by Ponizovskii, Putcha, and Ringel. In this paper, we present a similar result for the monoid of binary relations, a partial result for the…

In this note we show that unsatisfiable systems of linear equations with a constant number of variables per equation over prime finite fields have polynomial-size constant-degree semi-algebraic proofs of unsatisfiability. These are proofs…

This paper compares skew-linear and multilinear matroid representations. These are matroids that are representable over division rings and (roughly speaking) invertible matrices, respectively. The main tool is the von Staudt construction,…

Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or p-adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant,…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of…