Related papers: Unlabeled equivalence for matroids representable o…
Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…
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…
Over a finite field $\mathbb{F}_{q^m}$, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew…
In this paper we prove an identity in terms of generating functions which enables us to calculate the numbers of isomorphism classes of absolutely indecomposable semistable representations of quivers over finite fields.
A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory…
It is shown that the finite dimensional irreducible representaions of the quantum matrix algebra $ M_{ q,p}(2) $ ( the coordinate ring of $ GL_{q,p}(2) $) exist only when both q and p are roots of unity. In this case th e space of states…
We consider affine representable algebras, that is, finitely generated algebras over a field that can be embedded into some matrix algebra over a commutative algebra. We show that this algebra can in fact be chosen to be a polynomial…
In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More…
We study a certain family of finite-dimensional simple representations over quantum affine superalgebras associated to general linear Lie superalgebras, the so-called fundamental representations: the denominators of rational $R$-matrices…
We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic…
Let $M=(E,{\cal I})$ be a matroid. A {\em $k$-truncation} of $M$ is a matroid {$M'=(E,{\cal I}')$} such that for any $A\subseteq E$, $A\in {\cal I}'$ if and only if $|A|\leq k$ and $A\in {\cal I}$. Given a linear representation of $M$ we…
We conjecture that it is not possible to finitely axiomatize matroid representability in monadic second-order logic for matroids, and we describe some partial progress towards this conjecture. We present a collection of sentences in monadic…
We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group…
We compute the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. If the grading group is finite, we prove that the set of graded polynomial identities admits a finite basis. We…
We develop the rudiments of a finite-dimensional representation theory of groups over idempotent semifields by considering linear actions on tropical linear spaces. This can be considered a tropical representation theory, a characteristic…
If a $p$-adic Galois representation $\rho_{f,\nu}:\Gamma_{\mathbb Q} \to \GL_2(E_{f,\nu})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this…
We introduce the notion of the free product of $q$-matroids, which is the $q$-analogue of the free product of matroids. We study the properties of this noncommutative binary operation, making an extensive use of the theory of cyclic flats.…
We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the symmetry group consists of rotations and translations. Along the way, we use tropical…
A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…
We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n, p)$ is $(3, 3)$ or $(4,2)$. In the latter cases, we…