Hats, Blocks and Ribbons

Mike Zabrocki

Consider the space of symmetric functions \Lambda and the operators on this space (elements of Hom(\Lambda, \Lambda)). We introduce an involution which relates pairs of operators that add rows or columns to the partitions indexing symmetric functions (vertex operators).

A simple q-analog of this involution changes generating functions for Schur vertex operators to generating functions for the Poincare polynomials known as generalized Kostka polynomials. These generating functions allow us to compute explicit relations among the generalized Kostka polynomials that extend many of the relations that are known for the Kostka-Foulkes polynomials.

Finally, we use this involution to produce a combinatorial rule for the action of the operator that adds a column on the homogeneous symmetric functions when this operator acts on the Schur basis (a sort of dual to the Pieri rule since multiplcation by h_k adds a row).