Related papers: ${\rm SL}(3,\mathbb{C})$-character varieties of tw…

Via counting over finite fields, we derive explicit formulas for the E-polynomials and Euler characteristics of GL(d)- and PGL(d)-character varieties of free groups. We prove a positivity property for these polynomials and relate them to…

A local classification of semisimple algebras of vector fields on $\mathbb{C}^{3}$ is given, using the canonical forms of the Heisenberg algebra and of $sl(2,\mathbb{C})\times sl(2,\mathbb{C})$.

The modular transformation properties of admissible characters of the affine superalgebra sl(2|1;C) at fractional level k=1/u-1, u=2,3,... are presented. All modular invariants for u=2 and u=3 are calculated explicitly and an A-series and…

In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PSL(3;C) (in particular in PSL(2;C); PSL(3;R) and PU(2; 1)). The representations are obtained by gluing decorated tetrahedra of…

The first part of this article is a general introduction to the the theory of representation spaces of discrete groups into SL(n,C). Special attention is paid to knot groups. In Section 2 we discuss the difference between the tangent space…

We present the subalgebra structure of sl(3,O), a particular real form of e6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group SL(3,O) to…

This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a…

We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g \times 2g$ symplectic group for $L$ odd and $g \geq 3$.

We give a minimal system of 476 generators (resp. 510 generators) for the algebra of SL(2,C)-covariant polynomials on binary forms of degree 9 (resp. degree 10). These results were only known as conjectures so far. The computations rely on…

We categorify a quantized Heisenberg algebra associated to a finite subgroup of SL(2,C).

We determine the Galois group of the 2-class field tower for two particular families of imaginary quadratic number fields $k$ with $2$-class field tower of length $2$.

We construct new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\leq m\leq 3g-3$ non-intersecting simple loops on the Riemann surface and are…

We obtain explicit formulas for the product of a deformed Weyl denominator with the character of an irreducible representation of the spin group $\rm{Spin}_{2r+1}({\mathbb C})$, which is an analogue of the formulas of Tokuyama for Schur…

We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of $\text{\em SL}(2,\C)$-representation of the fundamental group. An essential input is a recent result of the second…

It has been shown recently that the toroidally compactified type IIB string effective action possesses an SL(2, R) invariance. Using this symmetry we construct an infinite family of macroscopic string-like solutions permuted by SL(2, Z)…

New finite-dimensional representations of specific polynomial deformations of sl(2,R) are constructed. The corresponding generators can be, in particular, realized through linear differential operators preserving a finite-dimensional…

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups…

In this note we give explicit isomorphisms of 2-categories between various versions of the categorified quantum group associated to a simply-laced Kac-Moody algebra. These isomorphisms are convenient when working with the categorified…

We describe the Stiefel-Whitney classes (SWCs) of orthogonal representations $\pi$ of the finite special linear groups $G=\text{SL}(2,\mathbb F_q)$, in terms of character values of $\pi$. From this calculation, we can answer interesting…

Let $\text{GL}(n) = \text{GL}(n, {\mathbb C})$ denote the complex general linear group and let $G \subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$). We…