Revisiting Brauer’s formula for tensor product decompositions

The notation used here will be explained below. A familiar (though oversimplified) example is given for the Lie algebra sl2(C) by the Clebsch– Gordan formula; as in [2, Exer. 22.7]. In 1937 Richard Brauer published a short note giving a general formula of this sort. It still serves as the starting point for some computer methods, even though it usually involves a large number of cancellations. Here our purpose is to revisit Brauer’s formula and related matters from the perspective of the BGG (Bernstein-Gelfand-Gelfand) category O attached to a semisimple Lie algebra g over an algebraically closed field (or other splitting field) of characteristic 0. These ideas from the early 1970s provide new insights into the finite dimensional Cartan–Weyl theory by working also with certain infinite dimensional modules (see [3] for a recent account). Our approach was suggested by J.C. Jantzen. He, along with Allen Knutson and Shrawan Kumar, also provided valuable comments on an early version of this note. It is difficult to say what might constitute the “simplest” or most transparent proof of Brauer’s formula, since by now the tensor product decomposition has been studied using many tools ranging from Lie algebra theory to algebraic geometry and combinatorics. Only a few references are included below. Apparently Brauer’s formula depends essentially on the Weyl character formula, but first we discuss some more elementary steps leading to qualitative estimates about the possible summands in the decomposition (∗).