arXiv research

Study representation rings and dimension functions for fusion systems.

New rings relate to Soergel categories, categorifying a representation.

Researchers explore representations of 3-manifold groups over finite commutative rings.

We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).

New Frobenius manifold structures found on Dicyclic group orbits.

Researchers found only one hyperbolic structure for Borromean rings.

The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.

Survey of recent developments in racks and quandles.

Study Coxeter groups over fusion rings and their geometric realisations.

The paper defines a ring structure in twisted equivariant $KK$-theory for Lie groups.

A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved and the Poisson K-rings are calcul…

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of $^*$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship betw…

Study on faithfulness of Burau representation for Artin-Tits groups.

Simpler equations derived for knot polynomials coefficients, forming a ring.

Researchers find Frobenius manifold structures on orbits spaces of finite groups.

New theory proves representability of PDE solutions without complex machinery.

The paper proves stabilization in hypersurface sections using Grothendieck rings and probabilistic methods.

Study SL(2,C) character schemes for finitely generated groups.

The paper classifies κ-twisted conjugacy classes and studies twining characters on Lie groups.

T-Basis represents neural network tensors with fewer parameters.

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representatio…

Researchers disprove Luo's conjecture on 3-manifold groups over commutative rings.

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.

FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian pro…

Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of such a ring defines new link invariants. In this sense, they produce the well…

Study shows Steinberg representation's multiplicity in cohomology of congruence subgroups.

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M can be expressed via these new integrals. The ring of exponential iterated inte…

Researchers lift knot coloring polynomial to Habiro ring.

We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).

Projective resolves symplectic Steinberg module for number rings.

Extends Lawrence's representations to integral $U_q \mathfrak{sl}(2)$ Verma-modules and braid groups.

Let M be a compact simply connected hyperkähler (or holomorphically symplectic) manifold, \dim H^2(M)=n. Assume that M is not a product of hyperkaehler manifolds. We prove that the Lie algebra so(n-3,3) acts by automorphisms on the cohomology ring H^*(M). Under this action, the space H^2(M) is isomorphic to the fundame…

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the \emph{algebra of symmetries} $\mathscr{S}(\Box^r)$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of highest weight repres…

Study on skein module dimensions at irreducible representations.

In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(Γ) of a compact Lie group $Γ$ to the complex K-theory of the classifying space $BΓ$. For infi…

New invariants for virtual knots and links defined via quiver representations.

New Alexander invariants for knot groups computed using $K_1$-groups.

New polynomial invariants for knots and links.

A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an $n\times n$ matrix with integer entries, or as a quiver in special cases, together with $n$ formal variables. A mutation is a c…

A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized …

In this paper we analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces (i.e. following Berezin, Leites, Manin, etc.). We establish a representation formula for all morphisms from the algebra of functions on an ordinary manifold…

Let $Γ$ be a finite d-valent graph and G an n-dimensional torus. An ``action'' of G on $Γ$ is defined by a map, $α$, which assigns to each oriented edge e of $Γ$ a one-dimensional representation of G (or, alternatively, a weight, $α_e$, in the weight lattice of G). For the assignment, $e \to α_e$, to be a schematic des…

Given any oriented link diagram, two types of new knot invariants are constructed. They satisfy some generalized skein relations. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of those rings define new link invariants. For example, the HOMFLYPT polynomial with three va…

Researchers compute K-theory for cohomogeneity-one actions.

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 $\mathbb{Q}[t,t^{-1}]$.

Proposes SPE for robust speaker verification.

New invariants defined for knots and links using quandle representations.

Let $M^n$ be a closed, connected $n$-manifold. Let $\mtm$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $\mtm$ is homotopy equivalent to the Spanier-Whitehead dual of $M$ with a disjoint basepoint, $M_+$. This dual can be viewed as the function spectrum, $F(M, S)$, whe…