Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let be a partition. A semistandard Young tableau of shape is a filling of the Young diagram of with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau is defined as where is the total number of times appears in the tableau.
where the sum is taken over all semistandard Young tableaux of shape .
As before we can readily verify that . This definition will allow us to deduce the Jacobi-Trudi identities for the Schur functions, which describe among other things the action of the fundamental involution . Since I’m trying to emphasize how many different ways there are to define the Schur functions, I’ll call these definitions instead of propositions.
Definition 5: .
Definition 6: .