Related papers: Noncommutative Independence in the Infinite Braid …
We construct a braided analogue of the quantum permutation group and show that it is the universal braided compact quantum group acting on a finite space in the category of $\mathbb{Z}/N\mathbb{Z}$-$\textrm{C}^*$-algebras with a twisted…
We introduce an infinite-dimensional affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by…
We give an upper bound on character sums of the resultant over pairs of monic square-free polynomials of given degrees, answering a question of Ellenberg and Shusterman in the quadratic case. Our approach is topological: we compute the…
We classify all the irreducible characters of a symmetric group such that the induced immanant function $d_{\chi}$ vanishes identically on alternate matrices with the entries in the complex field.
This is an expository article on the noncommutative singularity theory of power series in noncommuting variables, its motivation from deformation theory, and its applications to contractibility of curves and the classification of smooth…
The aim of the paper is to define noncommutative cluster structure on several algebras ${\mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a…
We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.
We take the fundamental group of the complement of the branch curve of a generic projection induced from canonical embedding of a surface. This group is stable on connected components of moduli spaces of surfaces. Since for many classes of…
After a quick review of the representation theory of the symmetric group, we give an exposition of the tools brought about by the so-called half-infinite wedge representation of the infinite symmetric group. We show how these can be applied…
We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of…
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…
Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically…
This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also…
We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S_3, the dihedral group D_4 and the quaternion group Q. Poincare' duality holds in every case, and under some…
We characterize unitary representations of braid groups $B_n$ of degree linear in $n$ and finite images of such representations of degree exponential in $n$.
We generalize presentations of the fundamental group of discriminant complements and arrive at a class of presentations associated naturally with words in the free monoid of the alphabet $\sigma_1,\dots,\sigma_{n-1}$. Our study addresses…
In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or…
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…
In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a…
General semifinite factor representations of the diffeomorphism group of euclidean space are constructed by means of a canonical correspondence with the finite factor representations of the inductive limit unitary group. This construction…