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A Pieri formula and a factorization formula for sums of $$K$$-theoretic $$K$$-Schur functions. (English) Zbl 1421.05096
Summary: We give a Pieri-type formula for the sum of $$K$$-$$k$$-Schur functions $$\sum _{\mu \le \lambda } g^{(k)}_{\mu }$$ over a principal order ideal of the poset of $$k$$-bounded partitions under the strong Bruhat order, whose sum we denote by $$\widetilde{g}^{(k)}_{\lambda }$$. As an application of this, we also give a $$k$$-rectangle factorization formula $$\widetilde{g}^{(k)}_{R_t\cup \lambda }=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda }$$ where $$R_t=(t^{k+1-t})$$, analogous to that of $$k$$-Schur functions $$s^{(k)}_{R_t\cup \lambda }=s^{(k)}_{R_t}s^{(k)}_{\lambda }$$.
MSC:
 05E15 Combinatorial aspects of groups and algebras (MSC2010) 05E05 Symmetric functions and generalizations 14N15 Classical problems, Schubert calculus 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20B30 Symmetric groups
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