A combinatorial description of the centralizer algebras connected to the Links–Gould invariant

Anghel, Cristina Ana-Maria

doi:10.2140/agt.2021.21.1553

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A combinatorial description of the centralizer algebras connected to the Links–Gould invariant

Cristina Ana-Maria Anghel

Algebraic & Geometric Topology 21 (2021) 1553–1593

DOI: 10.2140/agt.2021.21.1553

Abstract

We study the tensor powers of the standard representation of the super-quantum algebra $U_{q} (sl (2 | 1))$ , focusing on the rings of its algebra endomorphisms, called centralizer algebras and denoted by ${LG}_{n}$ . Their dimensions were conjectured by I Marin and E Wagner (Adv. Math. 248 (2013) 1332–1365). We prove this conjecture, describing the intertwiner spaces from a semisimple decomposition as sets consisting of certain paths in a planar lattice with integer coordinates. Using this model, we present a matrix unit basis for the centralizer algebra ${LG}_{n}$ , by means of closed curves in the plane, which are included in the lattice with integer coordinates.

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Keywords

quantum algebra, representation theory, centralizer algebras

Mathematical Subject Classification

Primary: 57K10

Secondary: 16T20, 17B37, 20F36, 57K31

References

Publication

Received: 3 July 2020

Revised: 16 November 2020

Accepted: 27 December 2020

Published: 11 August 2021

Authors

Cristina Ana-Maria Anghel
Mathematical Institute University of Oxford Oxford United Kingdom
https://www.maths.ox.ac.uk/people/cristina.palmer-anghel

Volume 21, issue 3 (2021)