We investigate a family of (reducible) representations of Artin's braid groups corresponding to a specific solution to the Yang-Baxter equation. The images of the braid groups under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of the braid groups factoring over Temperley-Lieb algebras and the corresponding link invariants.
Let E be the extraspecial p-group of order p^3 and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence.
Let E be the extraspecial p-group of order p^3 and exponent p where p is an odd prime. We consider the mod p cohomology of summands in the stable splitting of p-completed classifying space BE. Moreover, we consider the stable splitting for some finite groups with Sylow p-subgroup E.
This paper was motivated by a remarkable group, the maximal subgroup $M=S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}\ltimes2^{6+1}_{-}$ of the sporadic simple group ${\rm Fi}_{23}$, where $S_3$ is the symmetric group of degree 3, and $2^{2+1}_{-}$, $3^{2+1}$ and $2^{6+1}_{-}$ denote extraspecial groups. The representation $3^{2+1}\to{\rm GL}(3,\mathbb{F}_4)\to{\rm GL}(6,\mathbb{F}_2)$ extends (remarkably) to $S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}$ and preserves a quadratic form (of minus type) which allows the construction of $M$. The paper describes certain (Weil) representations of extraspecial group…
Weight conjectures for fusion systems on an extraspecial group
University of Manchester（曼彻斯特大学）；City St Georges, University of London（伦敦大学城市圣乔治学院）；University of Louisiana at Lafayette（路易斯安那拉斐特大学）；Loughborough University（拉夫堡大学）
We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. We define homological operators and we show that they actually provide a representation for an integral version for $U_q \mathfrak{sl}(2)$. We suggest an isomorphism between a given basis of homological modules and the standard basis of tensor products of Verma modules, and we show it to preserve the integral ring of coefficients, the action of $U_q \mathfrak{sl}(2)$, the braid group representations and their grading. This recovers an in…
We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the center of the braid group has finite order on the specialized Laurence representations, this action is faithful for our extension.
Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator $σ_1$ is between 2 and 5 the kernel projects onto a congruence subgroup of $\mathrm{PSL}(2,\mathbb{Z})$ and compute its level. However, we …
We construct a family of six dimensional block representations of the braid group $B_3$ on three strings. We show that some of these representations can be used to detect braid vertibility of some known knots and others of 9 and 10 crossings.
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed modular category $\mathcal{C}(A,Q)$ and that of $\mathcal{C}(A,Q)$ itself.
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated with quantum groups at roots of unity. We also show that in the case of p-groups, the corresponding pure braid group representations factor through a finite p-group, which answers a question asked of the first author by V. Drinfeld.
We establish strong constraints on the kernel of the (reduced) Burau representation $β_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm 1}\right]\right)$ of the braid group $B_4$. We develop a theory to explicitly determine the entries of the Burau matrices of braids in $B_4$, and this is an important step toward demonstrating that $β_4$ is faithful (a longstanding question posed in the 1930s). The theory is based on a novel combinatorial interpretation of $β_4\left(g\right)$, in terms of the Garside normal form of $g\in B_4$ and a new product decomposition of positive braids. We develop cance…
A braid representation is a monoidal functor from the braid category $\mathsf{B}$, for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category $\mathsf{C}$ with $ob(\mathsf{C})=\mathbb{N}$, a rank-$N$ charge-conserving representation (or spin-chain representation) is a strict monoidal functor $F$ from $\mathsf{C}$ to the category $\mathrm{Match}^N$ of rank-$N$ charge-conserving matrices that is natural in the sense that $F(1)=1$}. In this work we construct all spin-chain braid representations, and classify up to suitable notions of isomorphism.
In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equation to have irregular singularities. We also give a conjectural equivalence between these two classes of representations.
Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{Mat}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that …
The necklace braid group $\mathcal{NB}_n$ is the motion group of the $n+1$ component necklace link $\mathcal{L}_n$ in Euclidean $\mathbb{R}^3$. Here $\mathcal{L}_n$ consists of $n$ pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group $\mathcal{NB}_n$, especially those obtained as extensions of representations of the braid group $\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. We show that any irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in …
We study representations of the loop braid group $LB_n$ from the perspective of extending representations of the braid group $B_n$. We also pursue a generalization of the braid/Hecke/Temperlely-Lieb paradigm---uniform finite dimensional quotient algebras of the loop braid group algebras.
We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $SO(N)_2$ (for $N$ odd) and $O(N)_2$ (for $N$ even) in terms of quantum $(n-1)$-tori, via non-standard deformations of $U\mathfrak{so}_N$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image.
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 group. We give an exact formula for the first Betti number; it is a quartic polynomial in the number of strands. We also show that, like the pure braid group, the first homology satisfies uniform representation stability in the sense of Church and Farb. Unlike the pure braid group, the group of symmetries - the quotient of the braid group by the level 4 subg…
We show that the braid group representations associated with the $(3,6)$-quotients of the Hecke algebras factor over a finite group. This was known to experts going back to the 1980s, but a proof has never appeared in print. Our proof uses an unpublished quaternionic representation of the braid group due to Goldschmidt and Jones. Possible topological and categorical generalizations are discussed.
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 question of Graham Denham. We also use our result to compute the stable integral homology of the Milnor fiber.
We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general representations were given by I.Tuba and H.Wenzl [11]. They involve [(n+1)/2] parameters (and also u…
We show that Lawrence's representation and linear representations from quantum sl_2 called generic highest weight vectors detect the dual Garside length of braids in a simple and natural way. That is, by expressing a representation as a matrix over a Laurent polynomial ring using certain natural basis, the span of the variable is equal to the constant multiples of the dual Garside length.
The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful. We will also derive new presentations for the surface singular braid monoid, one with reduced the number of defining relations, and the other with reduced the number of its singular generators. We include surface singular braid formulations of all knotted surfaces in Yoshikawa'…
We list the irreducible two dimensional complex representations of the Braid group B3 in elementary way. Then, we make a decomposition of the square of its irreducible Burau representation.
We describe a new technique to obtain representations of the braid group B_n from the R-matrix of a quantum deformed algebra of the one dimensional harmonic oscillator. We consider the action of the R-matrix not on the tensor product of representations of the algebra, that in the harmonic oscillator case are infinite dimensional, but on the subspace of the tensor product corresponding to the lowest weight vectors.
In [1] we have constructed a [n+1/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension using a $q-$deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries (2000), who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand (2000) obtained an equivalent representation of B_3 by considering two special operators in the space ${\mathbb C}^n[X].$ Slightly more general representations were given by I. Tuba and H. Wenzl (2001). They involve [n…
Governed by locality, we explore a connection between unitary braid group representations associated to a unitary $R$-matrix and to a simple object in a unitary braided fusion category. Unitary $R$-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get rep…
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 representations are rigid if they are determined by the path algebra and the representations of $B_2$. We show that besides the known classical cases also the braid representations for the path algebra for the 7-dimensional representation of $G_2$ satisfies the rigidity condition, provided $B_3$ generates $\End(V^{\otimes 3})$. We obtain a complete classification …
The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the $n$-punctured disc. In this note, we calculate $H_*(B_n;V_n)$ as a module over the Laurent polynomial ring $\mathbb{Q}[t,t^{-1}]$.
We consider Albeverio- Rabanovich linear representation $π$ of the braid group $B_3$. After specializing the indeterminates used in defining the representation to non-zero complex numbers, we prove that the restriction of $π$ to the pure braid group $P_3$ of dimension three is irreducible.
We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.
In this note, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.
We give a method to construct new self-adjoint representations of the braid group. In particular, we give a family of irreducible self-adjoint representations of dimension arbitrarily large. Moreover we give sufficient conditions for a representation to be constructed with this method.
We give a method to produce representations of the braid group $B_n$ of $n-1$ generators ($n\leq \infty$). Moreover, we give sufficient conditions over a non unitary representation for being of this type. This method produces examples of irreducible representations of finite and infinite dimension.
In the present paper, we construct a variant of the Burau representation of two generalizations of the classical braid group. For the Gassner representation, we propose an iterative procedure to find and generalize the extension of this representation.
In this paper we determine the complex generic representation theory of the Juyumaya algebra. We show that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by P. Martin.
In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.
This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids, with the help of Drinfeld associators. We set a dictionary between representation-theoretic properties of these two structures, and tools to describe the representations thus obtained. We give an explanation for the frequent apparition of unitary structures on classical representations. We introduce new objects -- varieties of braided extensions, infinitesim…
The Long-Moody construction is a method to obtain representations of braid groups introduced by Long and Moody. Also the Katz middle convolution is known to be a method to construct local systems on $\mathbb{C}\backslash\{n\text{-points}\}$ introduced by Katz. In this paper, we explain that these two methods are naturally unified and define a new functor which we call the Katz-Long-Moody functor. This functor extends the framework of Katz algorithm to categories of local systems on various topological spaces, for example, $B_{n}$-bundles associated with simple Weierstrass polynomials, compleme…
For any n>3, we give a family of finite dimensional irreducible representations of the braid group B_n. Moreover, we give a subfamily parametrized by 0<m<n of dimension the combinatoric number (n,m). The representation obtained in the case m=1 is equivalent to the Standard representation.
The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(\mathfrak{sl}_2)$ are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the $q$-Racah polynomials which appear as $6j$-symbols in the Racah problem for $U_q(\mathfrak{sl}_2)$. Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group represen…
The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.
We construct two families of representations of the braid group $B_n$ by considering conjugation actions on congruence subgroups of $GL_{n-1}(Z[t^{\pm 1},q^{\pm 1}])$. We show that many of these representations are faithful modulo the center of $B_n$.
A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found. Following a rescaling and change of parameters, the matrices are found to be identical to those of the Krammer representation. The two representations are thus the same, proving the irreducibility of one and the faithfulness of the other.
We show that the Schur-Weyl type duality between $gl(1|1)$ and $GL_n$ gives a natural representation-theoretic setting for the relation between reduced and non-reduced Burau representations.
In a recent paper here arXiv:1508.0005 it is shown that irreducible representations of the three string braid group $B_3$ of dimensions $\leq 5$ extend to representations of the 3-component loop braid group $LB_3$. Further, an explicit $6$-dimensional irreducible $B_3$-representation is given that does not extend. In this note we give a necessary and sufficient condition, in all dimensions, on the components of irreducible representations of the modular group $Γ$ such that sufficiently general representations extend to $Γ\ast_{C_3} S_3$. As a consequence, the corresponding irreducible $B_3$-re…
In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$, $VB_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}, t_1^{\pm1},t_2^{\pm1},\ldots, t_{n-1}^{\pm1}]\right)$ which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful representations of crystallographic groups $B_n/P_n'$, $VB_n/VP_n'$, respectively. Using these representations we study certa…
We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.
The singularities of the representation variety of $B_3$ are studied, where $B_3$ is the knot group on 3 strands. Specifically, we determine which semisimple representations are smooth points of this variety.
In this paper we describe the structure of a group of conjugating automorphisms $C_n$ of free group and prove that this structure is similar to the structure of a braid group $B_n$ with $n>1$ strings. We find the linear representation of group $C_n$. Also we prove that the braid group $B_n(S^2)$ of 2--sphere, mapping class group M(0,n) of the $n$--punctured 2--sphere and the braid group $B_3(P^2)$ of the projective plane are linear. Using result of J. Dyer, E. Formanek, E. Grossman and the faithful linear representation of Lawrence--Krammer of $B_4$ we construct faithful linear representation …
Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view of proving the linearity of these groups. We decompose this representation in irreducible components and compute its Zariski closure, as well as its restriction to parabolic subgroups. We prove that it is faithful when W is a Coxeter group of type ADE and odd dihedral types, and conjectu…
As the 3-string braid group B(3) and the modular group PSL(2,Z) are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most' irreducible representations by constructing for each d-dimensional irreducible component X of the variety iss(n,B(3)) classifying the isomorphism classes of semi-simple n-dimensional representations of B(3) an explicit minimal etale rational map A^d --> X having a Zariski dense image. Such rational dense parametrizations were obtained for all components when n < …
We determine the Zariski closure of the representations of the braid groups that factorize through the Birman-Wenzl-Murakami algebra, for generic values of the parameters $α,s$. For $α,s$ of modulus 1 and close to 1, we prove that these representations are unitarizable, thus deducing the topological closure of the image when in addition $α,s$ are algebraically independent.
