A Pieri-type formula and a factorization formula for sums of $K$-$k$-Schur functions

Abstract

We give a Pieri-type formula for the sum of $K$-$k$-Schur functions ${\sum }_{\mu \le \lambda }{g}_{\mu }^{(k)}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by ${\tilde{g}}_{\lambda }^{(k)}$. As an application of this, we also give a $k$-rectangle factorization formula ${\tilde{g}}_{R_t\cup \lambda }^{(k)}={\tilde{g}}_{R_t}^{(k)}{\tilde{g}}_{\lambda }^{(k)}$ where $R_t=(t^{k+1-t})$, analogous to that of $k$-Schur functions $s_{R_t\cup \lambda }^{(k)}=s_{R_t}^{(k)}{s}_{\lambda }^{(k)}$.

33 pages, revised thoroughly, results unchanged