Multiplicity-Free Products of Schur Functions

We classify all multiplicity-free products of Schur functions and all multiplicity-free products of characters of SL(n; C). 0. Introduction In this paper, we classify the products of Schur functions that are multiplicity-free; i.e., products for which every coeecient in the resulting Schur function expansion is 0 or 1. We also solve the slightly more general classiication problem for Schur functions in any nite number of variables. The latter is equivalent to a classiication of all multiplicity-free tensor products of irreducible representations of GL(n) or SL(n). Multiplicity-free representations have many applications, typically based on the fact that their centralizer algebras are commutative, or that their irreducible decompositions are canonical; see the survey article by Howe H]. We nd it surprising that such a natural classiication problem seems not to have been considered before. Two well-known examples of multiplicity-free products are the Pieri rules (which correspond to a tensor product in which one of the factors is a symmetric or exterior power of the deening representation), and the rule for multiplying Schur functions of rectangular shape. The fact that the latter is multiplicity-free was rst noticed by Kostant, and has played an important role in several applications. For example, Stanley used the rule to count self-complementary plane partitions St]. More recently, the fact that these products are multiplicity-free has been a key property needed for explicit bijections constructed