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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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