Combinatorial aspects of generalizations of Schur functions

The understanding of the space of symmetric functions is gained through the study of its bases. Certain bases can be dened by purely combinatorial methods, sometimes enabling important properties of the functions to fall from carefully constructed combinatorial algorithms. A classic example is given by the Schur basis, made up of functions that can be dened using semi-standard Young tableaux. The Pieri rule for multiplying an important special case of Schur functions is proven using an insertion algorithm on tableaux that was dened by Robinson, Schensted, and Knuth. Furthermore, the transition matrices between Schur functions and other symmetric function bases are often linked to representation theoretic multiplicities. The description of these matrices can sometimes be given combinatorially as the enumeration of a set of objects such as tableaux. A similar combinatorial approach is applied here to a basis for the symmetric function space that is dual to the Grothendieck polynomial basis. These polynomials are dened combinatorially using reverse plane partitions. Bijecting reverse plane partitions with a subset of semi-standard Young tableaux over a doubly-sized alphabet enables the extension of RSK-insertion to reverse plane partitions. This insertion, paired with a sign changing involution, is used to give the desired combinatorial proof of the Pieri rule for this basis. Another basis of symmetric functions is given by the set of factorial Schur functions. While their expansion into Schur functions can be described combinatorially, the reverse change of basis had no such formulation. A new set of combinatorial objects is introduced to describe the expansion coecients, and another sign changing involution is used to prove that these do in fact encode the transition matrices.