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

Abstract

We study the tensor powers of the standard representation of the super-quantum algebra $U_q(\text{sl}(2|1))$, 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.