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

Takigiku, Motoki

Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 447-480.

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

Classification : 05E05, 20F55
Mots clés : $K$-theoretic $k$-Schur functions, Pieri rule, Coxeter groups, affine symmetric groups

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