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

TL;DR

The paper identifies the center of the Birman–Murakami–Wenzl algebras over $\mathbb{C}$ for generic parameters as the Wheel Laurent polynomials in the Jucys–Murphy elements, $WL[x_1,\dots,x_n]$, and develops an Okounkov–Vershik–style representation theory via a Murphy/GZ framework. It connects the BMW algebras to their affine counterparts to transfer center information, showing the Gelfand–Zeitlin algebra is generated by $x_i^{\pm1}$ and that central elements separate simples in the generic semisimple case. In non-generic regimes, notably non-semisimple even-power cases, the wheel subalgebra remains large enough to separate blocks in many instances, with an explicit admissibility–based description of blocks. The work further provides constructive descriptions of primitive idempotents in terms of JM data and branching paths, offering an explicit OV-like path to building irreducibles and illuminating the interplay between central structure and block decomposition in BMW algebras.

Abstract

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.

Theorems & Definitions (108)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Theorem 2.3
• Definition 3.1
• Example 3.2
• Example 3.3
• Example 3.4