Related papers: The virtual singular twin monoid and group: presen…

We pursue an analogy of the Schur-Weyl reciprocity for the spinor groups and pick up the irreducible spin representations in the tensor space $\Delta \textstyle{\bigotimes \bigotimes^k V}$. Here $\Delta$ is the fundamental representation of…

The twin group $TW_n$ on $n$ strands is the group generated by $t_1, \dots, t_{n-1}$ with defining relations $t_i^2=1$, $t_it_j = t_jt_i$ if $|i-j|>1$. We find a new instance of semisimple Schur--Weyl duality for tensor powers of a natural…

In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to…

We develop a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the…

Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel as broad generalizations of the well-known virtual braid groups. For each Coxeter graph $\Gamma$, they defined the virtual Artin group $VA[\Gamma]$, which is…

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a…

We study the construction of premonoidal categories, where the pentagon relation fails, through representations of finite group algebras and their quantum doubles. Both finite group algebras and their quantum doubles have a finite number of…

We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf-Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the…

By exploring simplicial structure of pure virtual braid groups, we give new connections between the homotopy groups of the 3-sphere and the virtual braid groups that are related to the theory of Brunnian virtual braids. The group structure…

We provide an (almost) self-contained construction of the Witten-Reshetikhin-Turaev representations of the mapping class group. We describe its properties including its Hermitian structure, irreducibility and integrality (at prime level).…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda} = (\lambda_{1}, \dots{} ,\lambda_{k})$ with $1 \le \lambda_{1} < \lambda_{2} < \dots{} < \lambda_{k}$, we…

We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations where all the defining relations are of the form $r=1$. We develop new approaches for finding…

We introduce the universal virtual braid group $UV_n(c)$, which provides a unified algebraic framework for virtual braid--type structures with $c$ types of crossings and admits natural quotient maps onto the standard families in the…

The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $\sigma_1$ and $\sigma_1 \sigma_2$ is known to yield an exotic Garside structure on $\mathcal{B}_3$. We introduce and study an infinite family $(M_n)_{n\geq 1}$ of…

In 1996 E. Formanek classified all the irreducible complex representations of $B_n$ of dimension at most $n-1,$ where $B_n$ is the Artin braid group on $n$ strings. In this paper we extend this classification to the representations of…

We show that the mirabolic quantum group $MU(n)$ is a comodule algebra over the quantized enveloping algebra $U_v(\mathfrak{sl}_n)$, and use this structure to give a complete classification of its finite dimensional representations. In…

Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group $G$ is split (or…

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…

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group $S_n$ has three natural extensions, namely, the braid group $B_n$,…

Twisted knot theory, introduced by M.O. Bourgoin, is a generalization of virtual knot theory. It naturally yields the notion of a twisted braid, which is closely related to the notion of a virtual braid due to Kauffman. In this paper, we…