Schur–Weyl-Type Duality for Quantized gl(1|1), the Burau Representation of Braid Groups, and Invariants of Tangled Graphs

Reshetikhin, Nicolai; Stroppel, Catharina; Webster, Ben

doi:10.1007/978-0-8176-8277-4_16

Cited by 8 publications

(9 citation statements)

References 11 publications

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“…Because of (2), we interpret Theorem 1 as a Schur-Weyl duality between U i .sl 2 / and the reduced SL 2 .C/-twisted Burau representation. This extends a similar result for U q .gl.1j1// and abelian SL 2 .C/-representations due to N. Reshetikhin, C. Stroppel, and B. Webster [23].…”

Section: Schur-weyl Duality For the Burau Representationsupporting

confidence: 87%

Holonomy invariants of links and nonabelian Reidemeister torsion

McPhail-Snyder¹

2022

Quantum Topol.

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We show that the reduced SL 2 .C/-twisted Burau representation can be obtained from the quantum group U q .sl 2 / for q D i a fourth root of unity and that representations of U q .sl 2 / satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the SL 2 .C/-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet-Geer-Patureau-Mirand-Reshetikhin, and we interpret their invariant as a twisted Conway potential.

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Section: Schur-weyl Duality For the Burau Representationsupporting

confidence: 87%

Holonomy invariants of links and nonabelian Reidemeister torsion

McPhail-Snyder¹

2022

Quantum Topol.

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“…In particular, if ∆ can be seen as a quantum invariant [23,25,30], it is in essence a classical invariant derived from a presentation matrix of the rst homology group of the innite cyclic covering of the complement of a given link in S 3 [1]. Therefore we can ask if some of ∆'s homological properties extend to LG.…”

Section: Introductionmentioning

confidence: 99%

The Links–Gould Invariant as a Classical Generalization of the Alexander Polynomial?

Kohli

2016

Experimental Mathematics

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In this paper we conjecture that the Links-Gould invariant of links, that we know is a generalization of the Alexander-Conway polynomial, shares some of its classical features. In particular it seems to give a lower bound for the genus of links and to provide a criterion for beredness of knots. We give some evidence for these two assumptions. Contents Introduction 1 1. The Links-Gould invariant and the genus of links 2 2. Evidence supporting the genus conjecture 5 3. The Links-Gould polynomial and beredness 15 4. Appendix : proof of the harder case in Theorem 2.23 17 References 20 2010 Mathematics Subject Classication. 57M27.

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“…• In the braid group/R-matrix formalism for U q (sl(N |M )) and U q (sl(1|1)) [25,34,35] in particular in [36], where it was also noticed that the various formulas for U q (sl(N |M )), like the values of Casimir invariants, skein relation, etc. looked as ones for U q (sl(N − M )).…”

Section: Perturbative Analysis Of the Alexander Polynomialmentioning

confidence: 99%

A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras

Mishnyakov,

Sleptsov,

Tselousov

2020

Preprint

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We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of N and generalizes the previously known "tug-the-hook" symmetry of the colored Alexander polynomial [1]. As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property imposes on the general group-theoretical structure, namely the sl(N ) weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture.

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Schur–Weyl-Type Duality for Quantized gl(1|1), the Burau Representation of Braid Groups, and Invariants of Tangled Graphs

Abstract: We show that the Schur-Weyl type duality between gl(1|1) and GLn gives a natural representation-theoretic setting for the relation between reduced and non-reduced Burau representations. %endquote

Cited by 8 publications

References 11 publications

Holonomy invariants of links and nonabelian Reidemeister torsion

Holonomy invariants of links and nonabelian Reidemeister torsion

The Links–Gould Invariant as a Classical Generalization of the Alexander Polynomial?

A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras

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