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On some factorization formulas of $$K$$-$$k$$-Schur functions. (English) Zbl 1397.05201
Summary: We give some new formulas about factorizations of $$K$$-$$k$$-Schur functions $$g^{(k)}_\lambda$$, analogous to the $$k$$-rectangle factorization formula $$s^{(k)}_{R_{t}\cup\lambda}=s^{(k)}_{R_{t}}s^{(k)}_{\lambda}$$ of $$k$$-Schur functions, where $$\lambda$$ is any $$k$$-bounded partition and $$R_t$$ denotes the partition $$(t^{k+1-t})$$ called a $$k$$-rectangle. Although a formula of the same form does not hold for $$K$$-$$k$$-Schur functions, we can prove that $$g^{(k}_{R_{t}}$$ divides $$g^{(k)}_{R_{t}\cup\lambda}$$, and in fact more generally that $$g^{(k)}_P$$ divides $$g^{(k)}_{P\cup\lambda}$$ for any multiple $$k$$-rectangles $$P$$ and any $$k$$-bounded partition $$\lambda$$. We give the factorization formula of such $$g^{(k)}_P$$ and explicit formulas for $$g^{(k)}_{P\cup\lambda}/g^{(k)}_P$$ in some cases.

##### MSC:
 05E05 Symmetric functions and generalizations 14N15 Classical problems, Schubert calculus
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##### References:
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