Volume 21, issue 3 (2021)

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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
Abstract

We study the tensor powers of the standard representation of the super-quantum algebra ${U}_{q}\left(sl\left(2|1\right)\right)$, 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