Tensor Product Representations for Orthosymplectic Lie Superalgebras

Abstract

We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to deene an action of the Brauer algebra on the tensor space V k which commutes with the action of the orthosymplectic Lie superalgebra spo(V) and the orthosymplectic Lie color algebra spo(V;). We use the Brauer algebra action to compute maximal vectors in V k and to decompose V k into a direct sum of submodules T. We compute the characters of the modules T , give a combinatorial description of these characters in terms of tableaux, and model the decomposition of V k into the submodules T with a Robinson-Schensted-Knuth type insertion scheme. Outline x0. Introduction (i) Summary of results (ii) Remarks on the results in this paper (iii) Open problems (iv) Acknowledgements x1. Lie color algebras and spo(V;) (i) Lie color algebras (ii) The category of nite dimensional modules for a Lie color algebra (iii) The general linear Lie color algebra gl(V;) (iv) The orthosymplectic Lie color algebras spo(V;) (v) Roots and root vectors in spo(V;) x2. The Brauer algebra action on tensor space (i) The unfolding map (ii) The Brauer algebra (iii) Some facts about braided rigid monoidal categories (iv) The braid group action on V k (v) The maps and (vi) The commuting action of the Brauer algebra on the spo(V;)-module V k (vii) spo(V;) invariants in V 2k x3. Maximal vectors in tensor space (i) Young symmetrizers, contractions, and spo(V;)-submodules of V k (ii) Construction and linear independence of maximal vectors x4. The spo(V;)-modules T and their characters (i) Deenition of the modules T (ii) Characters of the Brauer algebras (iii) Weighted traces (iv) Symmetric functions and a combinatorial description of char(T) x5. Tableaux and an insertion scheme (i) Up-down tableaux and spo-tableaux (ii) Punctured tableaux and the maps jeu and injeu (iii) Insertion of a letter into an spo-standard tableau (iv) Insertion of a word into an spo-standard tableau 0. Introduction Summary of results In this paper we show that there are orthosymplectic Lie superalgebra and orthosym-plectic Lie color algebra analogues of the results developed by Berele and Regev BR] and Sergeev Se] for general linear Lie superalgebras. Our work corresponds in the superalge-bra and color algebra setting to what Brauer did in Br] by extending to orthogonal and symplectic groups Schur's classical results for general linear groups. In his thesis Sh1] and a subsequent …

Cite this paper

@inproceedings{Benkart2007TensorPR, title={Tensor Product Representations for Orthosymplectic Lie Superalgebras}, author={Georgia Benkart and Chanyoung Lee Shader and Arun Ram}, year={2007} }