Quantum walled Brauer algebra: commuting families, Baxterization, and representations

Abstract

ABSTRACT. For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys-Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a "universal transfer matrix" that generates commuting elements of the algebra.

Supersymmetric polynomials and the center of the walled Brauer algebra

“…Note that it is equivalent to saying that the supersymmetric polynomials in L i 's generate the center of H Λ r,s (q, ρ). (3) Recently A. M. Semikhatov and I. Y. Tipunin introduced some elements {J(r) i ; 1 ≤ i ≤ r + s} having similar diagrammatic presentations to T (j) and U (k), which are also called Jucys-Murphy elements in [33,Section 2.4]. In particular, J(r) r+k coincides with U (k) for 1 ≤ k ≤ s under an isomorphism between their algebra qwB r,s and H C(q,ρ) r,s (q, ρ).…”