[Paper Review] An isomorphism Theorem for Yokonuma--Hecke algebras and applications to link invariants

Nicolas Jacon, Loïc Poulain d’Andecy|arXiv (Cornell University)|Jan 26, 2015

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper establishes an explicit isomorphism between the Yokonuma–Hecke algebra $Y_{d,n}$ and a direct sum of matrix algebras over tensor products of Iwahori–Hecke algebras of type $A$, enabling a complete classification of Markov traces and the construction of 3-variable link invariants. The isomorphism allows transfer of representation-theoretic and topological results from classical Hecke algebras to Yokonuma–Hecke algebras, proving that the resulting invariants for classical knots are topologically equivalent to the HOMFLYPT polynomial.

ABSTRACT

We develop several applications of the fact that the Yokonuma--Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori--Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma--Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism.

Motivation & Objective

• To establish a concrete isomorphism between the Yokonuma–Hecke algebra $Y_{d,n}$ and a direct sum of matrix algebras over tensor products of Iwahori–Hecke algebras of type $A$.
• To use this isomorphism to classify all Markov traces on $Y_{d,n}$ and derive explicit formulas for their parameters.
• To construct 3-variable polynomial invariants for framed and classical knots and links from these traces.
• To investigate the topological equivalence of the resulting invariants with the HOMFLYPT polynomial using the isomorphism.

Proposed method

• Explicitly construct an isomorphism between $Y_{d,n}$ and a direct sum of matrix algebras over tensor products of Iwahori–Hecke algebras of type $A$, indexed by compositions of $n$ with $d$ parts.
• Use the isomorphism to deduce the semisimple and modular representation theory of $Y_{d,n}$ from that of the Iwahori–Hecke algebras of type $A$.
• Define a symmetrizing form on $Y_{d,n}$ via the canonical symmetrizing form of the Iwahori–Hecke algebra of type $A$, and compute its Schur elements as products of Schur elements of type $A$ algebras.
• Classify all Markov traces on $Y_{d,n}$ by translating the classification from the Iwahori–Hecke algebra side, using the isomorphism to express parameters in terms of trace values on generators.
• Construct 3-variable link invariants $\mathrm{F}\Delta_{Y,S}$ as linear combinations of invariants $\mathrm{F}\Gamma^{ ho}_{Y,\mu^0}$ with coefficients $\alpha_{\mu^0}$ given by explicit formulas.
• Prove that the resulting invariants for classical links are topologically equivalent to the HOMFLYPT polynomial by showing their equivalence under the isomorphism and trace parameterization.

Experimental results

Research questions

• RQ1 How can the representation theory of the Yokonuma–Hecke algebra $Y_{d,n}$ be fully classified using its isomorphism to tensor products of Iwahori–Hecke algebras of type $A$?
• RQ2 What is the complete classification of all Markov traces on $Y_{d,n}$, and how can their parameters be explicitly computed?
• RQ3 Can 3-variable polynomial invariants for framed and classical links be constructed from these traces, and what are their properties?
• RQ4 Are the invariants obtained from $Y_{d,n}$ topologically equivalent to the HOMFLYPT polynomial for classical knots?
• RQ5 How does the isomorphism between $Y_{d,n}$ and matrix algebras over Iwahori–Hecke algebras of type $A$ facilitate the transfer of representation-theoretic and topological results?

Key findings

• The Yokonuma–Hecke algebra $Y_{d,n}$ is isomorphic to a direct sum of matrix algebras over tensor products of Iwahori–Hecke algebras of type $A$, indexed by compositions of $n$ with $d$ parts.
• All Markov traces on $Y_{d,n}$ are classified, and their parameters $\alpha_{\mu^0}$ are explicitly computed as $\alpha_{\mu^0} = \frac{D_S^{\vert\mu^0\vert-1}}{|S|^{\vert\mu^0\vert}}$ when $\mu^0_a = 0$ for all $a \notin S$, and zero otherwise.
• The 3-variable link invariants $\mathrm{F}\Delta_{Y,S}$ are constructed as linear combinations of invariants $\mathrm{F}\Gamma^{\sqrt{q}^{-1}}_{Y,\mu^0}$ with coefficients $\alpha_{\mu^0}$, where $u$ and $v$ are expressed in terms of $q$ and $\lambda_S$ via $u^2 = q$, $v = q-1$, and $\lambda_S = D_S / |S|$.
• The invariants for classical links obtained from $Y_{d,n}$ are topologically equivalent to the HOMFLYPT polynomial, as confirmed by the isomorphism and trace parameterization.
• The symmetrizing form on $Y_{d,n}$ is induced from the canonical symmetrizing form on the Iwahori–Hecke algebra of type $A$, and its Schur elements are products of Schur elements of type $A$ algebras.
• The decomposition of the Markov trace $\widetilde{\rho}_{S,n}$ into the basis of traces $\rho_{\mu^0,n}$ is explicitly given by $\mathrm{F}\Delta_{Y,S} = \sum_{\mu^0 \in \mathrm{Comp}_d^0} \alpha_{\mu^0} \mathrm{F}\Gamma^{\sqrt{q}^{-1}}_{Y,\mu^0}$, with $\alpha_{\mu^0}$ as defined in (72).