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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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