Symmetrizers and antisymmetrizers for the BMW algebra

TL;DR

This paper explicitly determines the coefficients of symmetrizer and antisymmetrizer elements in the Birman-Murakami-Wenzl algebra, which generate its minimal two-sided ideals and generalize similar structures in Hecke algebras.

Contribution

It provides explicit formulas for the coefficients of symmetrizers and antisymmetrizers in the BMW algebra's graphical basis, advancing understanding of its ideal structure.

Findings

Explicit coefficients of symmetrizers determined

Explicit coefficients of antisymmetrizers determined

Generalizes known structures from Hecke algebras

Abstract

Let $n\in\mathds{N}$ and $B_n(r,q)$ be the generic Birman-Murakami-Wenzl algebra with respect to indeterminants $r$ and $q$. It is known that $B_n(r,q)$ has two distinct linear representations generated by two central elements of $B_n(r,q)$ called the symmetrizer and antisymmetrizer of $B_n(r,q)$. These generate for $n\geq 3$ the only one dimensional two sided ideals of $B_n(r,q)$ and generalize the corresponding notion for Hecke algebras of type $A$. The main result in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of $B_n(r,q)$.