Related papers: Representing the Special Linear Group with Block U…

In this study, a novel feature coding method that exploits invariance for transformations represented by a finite group of orthogonal matrices is proposed. We prove that the group-invariant feature vector contains sufficient discriminative…

This work provides five explicit constructions of the exceptional Lie algebra $\mathfrak{e}_8$, based on its semisimple subalgebras of maximal rank. Each of these models is graded by an abelian group, namely, $\mathbb{Z}_4$, $\mathbb{Z}_5$,…

We explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and…

A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In…

Let $R$ be an associative ring with unity $1$ and consider that $2,k$ and $2k\in \mathbb{N}$ are invertible in $R$. For $m\geq 1$ denote by $UT_n(m,R)$ and $UT_{\infty}(m,R)$, the subgroups of $UT_n(R)$ and $UT_{\infty}(R)$ respectively,…

In this paper we generalize the idea of "essentially unique" representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal…

Using a new definition of rank for representations of semisimple groups sharp results are proved for the decay of matrix coefficients of unitary representations of two types of non-split $p$-adic simple algebraic groups of exceptional type.…

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique,…

We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a…

Semi-direct products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of…

We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective…

We show that MIF (mixed-identity-free) linear groups are sharply MIF and linearly MIF. Along the way we provide a self contained proof of the strong approximation theorem, and a new (probabilistic) variant of the super approximation…

We prove that all linear Lie groups satisfying the conditions listed in the title are finite extensions of commutative Lie groups.

We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential…

We study the complex irreducible representations of special linear, symplectic, orthogonal and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of…

The main purpose of this paper is providing a simple method to generate the matrices of irreducible representations because it is useful to reduce the computational time of solving the eigenvalue problems. The only information we need to…

The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. One is a generalization to coalgebras over a commutative ring (in which case the linear…

Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length…

Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie algebra $\mathfrak{g}$. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from $X$ to $\mathfrak{g}$. The irreducible…

The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special…