ABSTRACT
In this chapter, we study combinatorial objects called tableaux. Informally, a tableau is a filling of the cells in the diagram of an integer partition with labels that may be subject to certain ordering conditions. We use tableaux to give a combinatorial definition of Schur polynomials, which are examples of symmetric polynomials. The theory of symmetric polynomials nicely demonstrates the interplay between combinatorics and algebra. We give a brief introduction to this vast subject in this chapter, stressing bijective proofs throughout.
The reader may find it helpful at this point to review the basic definitions concerning integer partitions (see §2.8). Table 10.1 summarizes the notation used in this chapter to discuss integer partitions. In combinatorial arguments, we usually visualize the diagram dg(µ) as a collection of unit boxes, where (i, j) ∈ dg(µ) corresponds to the box in row i and column j. The conjugate partition µ′ is the partition whose diagram is obtained from dg(µ) by interchanging the roles of rows and columns.
