Related papers: Conjugacy classes of regular integer matrices

We study a closed differential form on the symmetric space of positive definite matrices, which is defined using the Pfaffian and is $\mathsf{GL}_{2n}(\mathbb{Z})$ invariant up to a sign. It gives rise to an infinite family of unstable…

We show that the weighted fusion category algebra of the principal 2-block $b_0$ of $\mathrm{GL}_2(q)$ is the quotient of the $q$-Schur algebra $\mathcal{S}_2(q)$ by its socle, for $q$ an odd prime power. As a consequence, we get a…

Let $q$ be a prime power, $G=GL_n(q)$ and let $U\leqslant G$ be the subgroup of (lower) unitriangular matrices in $G$. For a partition $\lambda$ of $n$ denote the corresponding unipotent Specht module over the complex field $\C$ for $G$ by…

In this paper we study Cartan subalgebras in general and special linear algebras over a field of positive characteristic. We determined the conjugacy classes of Cartan subalgebras under the general linear groups, and count the explicit…

The algebra of monodromy matrices for sl(n) trigonometric R-matrices is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product of L-operators.…

In this note we first study regular $\mathbb{Z}$-graded local rings. We characterize commutative noetherian regular $\mathbb{Z}$-graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated…

In 1999 V. Ivanov and S. Kerov observed that structure constants of algebras of conjugacy classes of symmetric groups $S_n$ admit a stabilization (in a non-obvious sense) as $n\to \infty$. We extend their construction to a class of pairs of…

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the…

We construct new families of U_q(gl_n)-modules by continuation from finite dimensional representations. Each such module is associated with a combinatorial object - admissible set of relations defined in \cite{FRZ}. More precisely, we prove…

We consider the possible covariant external algebra structures for Cartan's 1-forms on GL_q(N) and SL_q(N). We base upon the following natural postulates: 1. the invariant 1-forms realize an adjoint representation of quantum group; 2. all…

Let $G$ be a finite group and $N<G$ a normal subgroup with $G/N$ abelian. We show how the conjugacy classes of $G$ in a given coset $qN$ relate to the irreducible characters of $G$ that are not identically $0$ on $qN$. We describe several…

We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the…

A generalized numerical semigroup is a submonoid of $\mathbb{N}^d$ with finite complement in it. In this work we study some properties of three different classes of generalized numerical semigroups. In particular, we prove that the first…

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…

Generalised algebraic theories (GATs) allow multiple sorts indexed over each other. For example, the theories of categories or Martin-L{\"o}f type theories form GATs. Categories have two sorts, objects and morphisms, and the latter are…

Let G be a finite subgroup of GL_n(C). A study is made of the ways in which resolutions of the quotient space C^n / G can parametrise G-constellations, that is, G-regular finite length sheaves. These generalise G-clusters, which are used in…

The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups…

Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\phi(z)=z^n+c_{n-1} z^{n-1}+\dots+c_0$ over $D$ a "companion polynomial" $\Phi(z)$ of degree $n d$ with coefficients in $F$ whose…

We compute the canonical integrals associated to wheel graphs, and prove that they are proportional to odd zeta values. From this we deduce that wheel classes define explicit non-zero classes in: the locally finite homology of the general…

Let $F$ be an infinite field and $UT(d_1,\dots, d_n)$ be the algebra of upper block-triangular matrices over $F$. In this paper we describe a basis for the $G$-graded polynomial identities of $UT(d_1,\dots, d_n)$, with an elementary grading…