A generalization of the Murnaghan-Nakayama rule for $K$-$k$-Schur and $k$-Schur functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914787549511680 |
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| author | Duc, Khanh Nguyen |
| author_facet | Duc, Khanh Nguyen |
| contents | The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal{F}_λ^{(k)}$, that generalizes the constructions via the Pieri rule of $K$-$k$-Schur functions and $ k$-Schur functions. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases and connections promise to be detailed in the future. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_02037 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A generalization of the Murnaghan-Nakayama rule for $K$-$k$-Schur and $k$-Schur functions Duc, Khanh Nguyen Representation Theory Combinatorics K-Theory and Homology 05E05, 14N15 The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal{F}_λ^{(k)}$, that generalizes the constructions via the Pieri rule of $K$-$k$-Schur functions and $ k$-Schur functions. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases and connections promise to be detailed in the future. |
| title | A generalization of the Murnaghan-Nakayama rule for $K$-$k$-Schur and $k$-Schur functions |
| topic | Representation Theory Combinatorics K-Theory and Homology 05E05, 14N15 |
| url | https://arxiv.org/abs/2212.02037 |