## Hats, Blocks and Ribbons

**Mike Zabrocki**

(UQAM)

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).