Related papers: Cabling Burau Representation
We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Simona Settepenella to representation stability in the sense of Church--Ellenberg--Farb, answering a…
We provide a complete classification of the homogeneous $3$-local representations of the twin group $T_n$, the virtual twin group $VT_n$, and the welded twin group $WT_n$, for all $n\geq 4$. Beyond this classification, we examine the main…
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…
Let $C_n$ be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation $\rho$ of $C_n$, which is an extension of Lawrence-Krammer representation of the braid group $B_n$. Bardakov proved that the representation…
We prove that braid group representations associated to braided fusion categories and mapping class group representations associated to modular fusion categories are always semisimple. The proof relies on the theory of extensions in…
We investigate the cohomology of the level 4 subgroup of the braid group, namely, the kernel of the mod 4 reduction of the Burau representation at $t=-1$. This group is also equal to the kernel of the mod 2 abelianization of the pure braid…
In this article we prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if we know presentation of virtual pure braid group $VP_4$, then we can find presentation of $VP_n$ for arbitrary $n > 4$. Using…
The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation $B_n \to \operatorname{GL}_{n}(\mathbb Z)$. We find the image of such congruence subgroups in $\operatorname{GL}_{n}(\mathbb…
In this paper, we study the representations of the triplet group $L_n$, where $n$ is a positive integer, and its extensions to the virtual and welded triplet groups $VL_n$ and $WL_n$, respectively. We first introduce $L_n$, its extensions,…
We study the Gassner representation of the pure braid group $P_n$ by considering its restriction to a free subgroup $F$. The kernel of the restriction is shown to lie in the subgroup $[\Gamma^3 F,\Gamma^2 F]$, sharpening a result of…
This paper aims to generalize Artin's ideas to establish an one-to-one correspondence between the orbit braid group $B^{orb}_n(\mathbb{C},\mathbb{Z}_p)$ and a quotient of a group formed by some particular homeomorphisms of a punctured…
Many topological phenomena first proposed and observed in the context of electrons in solids have recently found counterparts in photonic and acoustic systems. In this work, we demonstrate that non-Abelian Berry phases can arise when…
We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.
Exchanging particles on graphs, or more concretely on networks of quantum wires, has been proposed as a means to perform fault tolerant quantum computation. This was inspired by braiding of anyons in planar systems. However, exchanges on a…
Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $n\in \N$. We say that such…
Non-trivial braid-group representations appear as non-Abelian quantum statistics of emergent Majorana zero modes in one and two-dimensional topological superconductors. Here, we generate such representations with topologically protected…
We obtain a presentation for the singular part of the Brauer monoid with respect to an irreducible system of generators, consisting of idempotents. As an application of this result we get a new construction of the symmetric group via…
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.
A new method for deriving universal \v{R} matrices from braid group representation is discussed. In this case, universal \v{R} operators can be defined and expressed in terms of products of braid group generators. The advantage of this…
We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants…