The center of the BMW algebras and an Okounkov-Vershik like approach

We use the Jucys-Murphy elements of the BMW algebra to show that its center over the complex numbers for almost all parameters making it semisimple is given by Wheel Laurent polynomials, a subalgebra of the symmetric Laurent polynomials in the JM elements. As an application, we give an Okounkov-Vershik like approach to its finite dimensional representations. In the non semisimple case related to the type B Lie algebras, the central subalgebra of Wheel Laurent polynomials is large enough to separate blocks of the BMW algebras.

💡 Research Summary

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The paper investigates the centre of the Birman‑Murakami‑Wenzl (BMW) algebras (B_n(q,t)) using Jucys‑Murphy (JM) elements and develops an Okounkov‑Vershik‑type approach to their finite‑dimensional representation theory.

First, the authors recall the classical Schur‑Weyl duality for (\mathfrak{sl}_n) and the symmetric group, where JM elements generate a maximal commutative subalgebra of (\mathbb{C}S_n) and the centre is the algebra of symmetric polynomials in these elements. Analogous results are known for Brauer algebras, partition algebras, and various quantum analogues, where the centre is described by (super)symmetric polynomials satisfying an additive cancellation condition.

The BMW algebra (B_n(q,t)) is a two‑parameter deformation of the Brauer algebra, equipped with a natural tower (B_1\subset B_2\subset\cdots). The authors define JM elements (x_1,\dots,x_n) recursively; they commute with the lower algebras in the tower and generate a maximal commutative subalgebra analogous to a Cartan subalgebra.

To describe the centre, the paper introduces the Wheel Laurent polynomials: \