Remark on the Branching theorem and supersymmetric algebras

Supersymmetric algebras have already proved useful in giving trans- parent proofs of a number of basic results of representation theory [3, 7-9, 1 l-l 51. Specifically, the technique of introducing virtual variables, which may have a different signature than the signature of variables to be delt with, often cuts down the amount of computation. Furthermore, the extension of results of representation theory to the superalgebraic setting sheds new light, and permits us to establish natural correspondences that were formally missing. In this note we carry out this program by deriving a superalgebraic version of the Branching Rules for the representations of the general linear group. While the statement of supersymmetric branching rules is in all respects similar to the ordinary one (and differing from it in our allowing variables of two signature), the proof yields a useful dividend, namely, a simple combinatorial construction of a canonical basis for the decomposi- tion of a restriction of a representation. As an application we give a supersymmetric generalization of Pieri’s formula, as well as a proof of this formula which is perhaps as short as it can be whittled down to. This application has been inspired by some recent work of Bofli [S, 61. We have benefited from the pioneering work of Berele and Regev [3], who were first to state such a supersymmetric extension of branching rules, as well as from the insights of Balentekin and Bars [2]. 255