Related papers: On stated $SL(n)$-skein modules

We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor…

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-characters…

The author defined for each (commutative) Frobenius algebra a skein module of surfaces in a $3$-manifold $M$ bounding a closed $1$-manifold $\alpha \subset \partial M$. The surface components are colored by elements of the Frobenius…

In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to…

The Lie module of the group algebra $FS_n$ of the symmetric group is known to be not projective if and only if the characteristic $p$ of $F$ divides $n$. We show that in this case its non-projective summands belong to the principal block of…

Let $S$ be a closed, connected, orientable surface of genus at least 3, $\mathcal{C}(S)$ be the complex of curves on $S$ and $Mod_S^*$ be the extended mapping class group of $S$. We prove that a simplicial map, $\lambda: \mathcal{C}(S) \to…

In this paper we study the skein modules of the surfaces, $\Sigma_{i,j}$ $(i,j)\in \{(0,2),(0,3),(1,0),(1,1)\}$ at $2N$-th roots of unity where $N\geq 3$ is an odd counting number and construct Frobenius algebras from them.

Let $R_n$ denote the KLR algebra of type $A^{(1)}_{e-1}$. Using the presentation of Specht modules given by Kleschev-Mathas-Ram, Loubert completely determined $\hom_{R_n}(S^\mu,S^\lambda)$ where $\mu$ is an arbitrary partition, $\lambda$ is…

When the quantum parameter $q^{\frac{1}{2}}$ is a root of unity of odd order and the punctured bordered surface has nonempty boundary, we prove the fraction ring of the stated skein algebra (that is the localization over all nonzero…

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ providing a…

We obtain a family of skein identities in the Kauffman bracket skein module which relate Frobenius elements to Jones-Wenzl projectors at roots of unity. We view these skein identities as certain incarnations of Steinberg tensor product…

It is well known that $n$-dimensional projective group gives rise to a non-homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known…

The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures, $\mathfrak{S}=\Sigma_{g,m}$, is the quotient of the Kauffman bracket skein algebra $\mathcal{S}_\xi(\mathfrak{S})$ corresponding to fixing the scalar values of…

Let $M$ be a compact orientable 3-manifold. The set of characters of $SL_2({\mathbb C})$ representations of the fundamental group of $M$ forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization…

We prove a generalised version of finiteness of skein modules for 3-manifolds by including boundary. We show that internal skein modules are holonomic modules over the internal skein algebra of the boundary - a property including finite…

The Kauffman bracket skein modules, S(M,A), have been calculated for A=+1,-1, for all 3-manifolds M by relating them to the SL(2,C)-character varieties. We extend this description to the case when A is a 4-th root of 1 and M is either a…

The purpose of this paper is to answer a question raised by Gennady Lyubeznik and Karen Smith. This question involves the finite generation of the following non-commutative algebra. Let $S$ be any commutative algebra of prime characteristic…

Skein algebras of surfaces quantize character varieties of topological surfaces, and in low genus, these quantizations are often related to algebras arising in representation theory. For example, Terwilliger defined a universal $SL_2$…

We give an $SL_3$ analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Le. To any punctured bordered surface, we associate an $SL_3$ stated skein algebra which contains the $SL_3$ skein algebra…

We provide geometric constructions of modules over the graded Cherednik algebra $\mathfrak{H}^{gr}_\nu$ and the rational Cherednik algebra $\mathfrak{H}^{rat}_\nu$ attached to a simple algebraic group $\mathbb{G}$ together with a pinned…