The Theory of Schur Polynomials Revisited

where the sum is over all semistandard Young tableaux T of shape λ, and c(T ) denotes the content vector of T . The traditional approach to the theory of Schur polynomials begins with the classical definition (1); see for example [FH, M, Ma]. Since equation (1) is a special case of the Weyl character formula, this method is particularly suitable for applications to representation theory. The more combinatorial treatments [Sa, Sta] use (3) as the definition of sλ(x), and proceed from there. It is not hard to relate formulas (1) and (3) to each other directly; see e.g. [Pr, Ste]. In this article, we take the Jacobi-Trudi formula (2) as the starting point, where the hr represent algebraically independent variables. We avoid the use of the x variables or ‘alphabets’ and try to prove as much as we can without them. For this purpose, it turns out to be very useful to express (2) in the alternative form