Related papers: Cubic graphical regular representations of $PSL_2(…

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of…

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these…

Let $\mathcal{S}$ be a finite thick generalized quadrangle, and suppose that $G$ is an automorphism group of $\mathcal{S}$. If $G$ acts primitively on both the points and lines of $\mathcal{S}$, then it is known that $G$ must be almost…

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special…

The Prime Graph Question for integral group rings asks if it is true that if the normalized unit group of the integral group ring of a finite group $G$ contains an element of order $pq$, for some primes $p$ and $q$, also $G$ contains an…

Given a finite group $G$ and a conjugacy class of involutions $X$ of $G$, we define the commuting involution graph $\mathcal{C}(G,X)$ to be the graph with vertex set $X$ and $x,y \in X$ adjacent if and only if $x \neq y$ and $xy =yx$. In…

We extend fundamental results of small cancellation theory to groups whose presentations satisfy the generalizations of the classical C(6) and C(7) conditions in graphical small cancellation theory. Using these graphical small cancellation…

We study and classify almost all quantum SL(3,C)'s whose representation theory is ``similar'' to that of the (ordinary) group SL(3,C). Only one case, related to smooth elliptic curves, could not be treated completely.

We give a proof, using so-called fusion rings and q-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(\infinity)). This is obtained here as a…

For each finite primitive subgroup $G$ of $\operatorname{PGL}_4(\mathbb{C})$, we find all the smooth $G$-invariant quartic surfaces. We also find all the faithful representations in $\operatorname{PGL}_4(\mathbb{C})$ of the smooth quartic…

The quantum superalgebra $U_q[gl(2/1)]$ is given as both a Drinfel'd--Jimbo deformation of $U[gl(2/1)]$ and a Hopf superalgebra. Finite--dimensional representations of this quantum superalgebra are constructed and investigated in a basis of…

Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been studied for many years and the theory is well-developed, including generating sets for the asscoiated groups. This paper studies the properties of self-duality,…

We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational…

Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $\Pi$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus \Pi$ and two vertices adjacent if they either span a secant to…

It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general,…

Let $F$ be an algebraically closed field of characteristic $p$. We fashion an infinite dimensional basic algebra $\underleftarrow{\mathcal{C}}_p(F)$, with a transparent combinatorial structure, which we expect to control the rational…

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, \mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, \mathbb{F}_{7})$. We obtain a necessary and…

We study the finite-dimensional continuous complex representations of $\mathrm{SL}_2$ over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of…

The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a…

We prove that for each sufficiently complicated orientable surface $S$, there exists an infinite image linear representation $\rho$ of $\pi_1(S)$ such that if $\gamma\in\pi_1(S)$ is freely homotopic to a simple closed curve on $S$, then…