Related papers: Representations of algebraic groups over a 2-dimen…

Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups:…

Let $F$ be an infinite division ring, $V$ be a left $F$-vector space, $r>0$ be an integer. We study the structure of the representation of the linear group $\mathrm{GL}_F(V)$ in the vector space of formal finite linear combinations of…

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad…

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho \colon G(R)^+ \to GL_n (K)$ over an…

In this article we define $G$-algebras, that is, graded algebras on which a reductive group $G$ acts as gradation preserving automorphisms. Starting from a finite dimensional $G$-module $V$ and the polynomial ring $\mathbb{C}[V]$, it is…

Motivated by the study of the interrelation between functorial and algebraic quantum field theory, we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of…

Let G be a connected reductive group over a field of characteristic zero, and consider an orthogonal representation of G. We give a simple criterion for whether the representation lifts to the spin group, in terms of the highest weights of…

A linear algebraic group G is over a field K is called a Cayley K-group if it admits a Cayley map, i.e., a G-equivariant K-birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups…

Unitary Representations corresponding to local shifts in reference frames are a key topic for progress in Quantum Gravity. The fibre bundle construct defined in our previous work in which quantum fields become liftings of; or sections…

We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real…

We give an interpretation of the double affine Hecke algebra of Cherednik as the (suitably regularized) algebra of double cosets of a group G by a subgroup J, extending the well known interpretations of finite and affine Hecke algebras. In…

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear…

Let k be an algebraically closed field of characteristic >2, F=k((t)) and Mp(F) denote the metaplectic extension of Sp_{2d}(F). In this paper we propose a geometric analog of the Weil representation of Mp(F). This is a category of certain…

Let $F$ be a $\delta-$field (differential field) of characteristic zero with an algebraically closed field of constants $F^\delta$, $A$ be a $\delta-F-$central simple algebra, $K$ be a Picard-Vessiot extension for the $\delta-F-$module $A$…

A representation field for a non-maximal order H in a central simple algebra A is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we…

We define a new notion of cuspidality for representations of $\GL_n$ over a finite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedean local field $F$ using geometric and infinitesimal induction functors, which involve…

We study the representations of the group $\mathbb{Z}_2^{*n}$, the free product of $\mathbb{Z}_2$ with itself $n$-times. We use the action of $B_n = S_2 \wr S_n $ as algebra automorphisms on the group algebra $\mathbb{C}(\mathbb{Z}_2^{*n})$…

In this note we consider representations of the group GL(n,F), where F is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field F.…

Let $G$ be a group and let $\rho\colon G\to\operatorname{Sym}(V)$ be a permutation representation of $G$ on a set $V$. We prove that there is a faithful $G$-coalgebra $C$ such that $G$ arises as the image of the restriction of…

We introduce a new approach to representation theory of finite groups that uses some basic algebraic geometry and allows to do all the theory without using characters. With this approach, to any finite group $G$ we associate a finite number…