Bases for infinite dimensional simple $\mathfrak{osp}(1|2n)$-modules respecting the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$

TL;DR

This paper develops explicit combinatorial and algebraic bases for certain infinite-dimensional modules of rak{osp}(1|2n) by analyzing their branching to rak{gl}(n), providing new tools for their representation theory.

Contribution

It introduces new bases and explicit raising/lowering operators for rak{osp}(1|2n)-modules respecting the rak{gl}(n) branching, combining combinatorial and algebraic methods.

Findings

Constructed a new basis for modules using Young tableaux.

Derived explicit raising and lowering operators via extremal projectors.

Connected Gel'fand-Zetlin basis to the new basis through a triangular transition.

Abstract

We study the effects of the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$ on a particular class of simple infinite-dimensional $\mathfrak{osp}(1|2n)$-modules $L(p)$ characterized by a positive integer $p$. In the first part we use combinatorial methods such as Young tableaux and Young subgroups to construct a new basis for $L(p)$ that respects this branching and we express the basis elements explicitly in two distinct ways. First as monomials of negative root vectors of $\mathfrak{gl}(n)$ acting on the $\mathfrak{gl}(n)$-highest weight vectors in $L(p)$ and then as polynomials in the generators of $\mathfrak{osp}(1|2n)$ acting on the $\mathfrak{osp}(1|2n)$-lowest weight vector in $L(p)$. In the second part we use extremal projectors and the theory of Mickelsson-Zhelobenko algebras to give new explicit constructions of raising and lowering operators related to the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$. We use the raising operators to give new expressions for the elements of the Gel'fand-Zetlin basis for $L(p)$ as monomials of operators from $U(\mathfrak{osp}(1|2n))$ acting on the $\mathfrak{osp}(1|2n)$-lowest weight vector in $L(p)$. We observe that the Gel'fand-Zetlin basis for $L(p)$ is related to the basis constructed earlier in the paper by a triangular transition matrix. We end the paper with a detailed example treating the case $n=3$.