Symmetric polynomials

In this section, we investigate the fundamentals of the theory of symmetric polynomials, which are homogeneous polynomials invariant under permutations of the variables. The basic reference is Macdonalds' monograph [83] (see also [84]). Other important references are: Stanley's book [113], which is the best reference for the combinatorial aspects, and Sagan's [108], where the connections with the representation theory of the symmetric group are treated in great detail. We have also benefited greatly from the expositions in Fulton's and Fulton and Harris monographs [42, 43] (we adopt their point of view focusing on symmetric polynomials in a finite number of variables), and Simon's book [111], especially for the proof of the Frobenius character formula, that indeed will be given in the next subsection. We also mention the recent book by Procesi [103] and [46] which is a very nice historical survey, from Cauchy's formula to Macdonald polynomials. For particular aspects and applications of symmetric functions, we also refer to the books by Lascoux [77], Manivel [85] and Bressoud [12].

More notation and results on partitions

In this section, we collect some notation and results on partitions that will be frequently used in this chapter. Let n be a positive integer and λ = (λ1, λ2, …, λk) ⊢ n be a partition.