Diamond module for the Lie algebra $\mathfrak{so}(2n+1,\mathbb C)$

TL;DR

This paper extends the combinatorial diamond cone basis construction to the Lie algebra fso(2n+1), using orthogonal quasistandard Young tableaux, providing a new basis for the associated diamond module.

Contribution

It generalizes the diamond cone basis construction to fso(2n+1) using orthogonal quasistandard Young tableaux, filling a gap in the combinatorial representation theory.

Findings

Defined orthogonal quasistandard Young tableaux.

Proved these tableaux form a basis for the diamond module.

Extended the construction to fso(2n+1) from previous cases.

Abstract

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor $\mathfrak n$ of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for $\mathfrak{sl}(n)$, the rank 2 semi-simple Lie algebras and $\mathfrak{sp}(2n)$. In the present work, we generalize these constructions to the Lie algebras $\mathfrak{so}(2n+1)$. The orthogonal semistandard Young tableaux were defined by M. Kashiwara and T. Nakashima, they form a basis for the shape algebra of $\mathfrak{so}(2n+1)$. Defining the notion of orthogonal quasistandard Young tableaux, we prove these tableaux give a basis for the diamond module for $\mathfrak{so}(2n+1)$.