Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIdempotents of the Hecke algebra become Schur functions in the skein of the annulus
326
0
0.0
(
0
)
Authors
Sascha G. Lukac
Ask ChatGPT about the research
No Arabic abstract
The Hecke algebra H_n contains well known idempotents E_{lambda} which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_{lambda} was given by Aiston and Morton. The closure of E_{lambda} becomes an element Q_{lambda} of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s_{lambda}. But previous proofs of this fact used results about quantum groups which were far beyond the scope of skein theory. Our elementary proof uses only skein theory and basic algebra.
rate research
Read More
We introduce non-acyclic $PGL_n(mathbb{C})$-torsion of a 3-manifold with toroidal boundary as an extension of J. Portis $PGL_2(mathbb{C})$-torsion, and present an explicit formula of the $PGL_n(mathbb{C})$-torsion of a mapping torus for a surface with punctures, by using the higher Teichm{u}ler theory due to V. Fock and A. Goncharov. Our formula gives a concrete rational function which represents the torsion function and comes from a concrete cluster transformation associated with the mapping class.
Let $k$ be a subring of the field of rational functions in $x, v, s$ which contains $x^{pm 1}, v^{pm 1}, s^{pm 1}$. If $M$ is an oriented 3-manifold, let $S(M)$ denote the Homflypt skein module of $M$ over $k$. This is the free $k$-module generated by isotopy classes of framed oriented links in $M$ quotiented by the Homflypt skein relations: (1) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$; (2) $L$ with a positive twist $=(xv^{-1})L$; (3) $Lsqcup O=(frac{v-v^{-1}}{s-s^{-1}})L$ where $O$ is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of $S(S^1times D^2)$ given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for $S(S^1times D^2)$ based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements $s^{2n}-1$, for $n$ a nonzero integer, and the elements $s^{2m}-v^{2}$, for any integer $m$, are invertible in $k$, then $S(S^{1} times S^2)=k$-torsion module $oplus k$. Here the free part is generated by the empty link $phi$. In addition, if the elements $s^{2m}-v^{4}$, for $m$ an integer, are invertible in $k$, then $S(S^{1} times S^2)$ has no torsion. We also obtain some results for more general $k$.
We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y, and use it to give a concrete isomorphism between Y and Shojis modified Ariki-Koike algebra. We give a cellular basis for Y and show that the Jucys-Murphy elements for Y are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi-Juyumaya algebra of braids and ties.
Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over $S^2$.
Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $bar S^p$. It is shown that $S^p$ is a standardly based algebra and $bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
