A combinatorial description of the centralizer algebras connected to the Links-Gould Invariant

TL;DR

This paper proves a conjecture about the structure and dimensions of centralizer algebras related to the Links-Gould invariant, using a combinatorial lattice path model and planar diagrams.

Contribution

It provides a combinatorial description and a matrix unit basis for the centralizer algebras associated with the quantum super-algebra U_q(sl(2|1)).

Findings

Confirmed the conjectured dimensions of the centralizer algebras.

Described intertwiner spaces using paths in a planar lattice.

Constructed a basis of matrix units via closed curves in the plane.

Abstract

In this paper we study the tensor powers of the standard representation of the quantum super-algebra $U_q(sl(2|1)$, focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by $LG_n$. Their dimensions were conjectured by I. Marin and E. Wagner \cite{MW}. We prove this conjecture, describing the intertwiner spaces from a semi-simple 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 centraliser algebra $LG_n$, by means of closed curves in the plane, which are included in the lattice with integer coordinates.