Distinguished filtrations of the $0$-Hecke modules for dual immaculate quasisymmetric functions
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| Format: | Preprint |
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2025
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| author | Lee, So-Yeon Oh, Young-Tak |
| author_facet | Lee, So-Yeon Oh, Young-Tak |
| contents | Let $α$ range over the set of compositions. Dual immaculate quasisymmetric functions $\mathfrak{S}_α^*$, introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki, provide a quasisymmetric analogue of Schur functions. They also constructed an indecomposable $0$-Hecke module $\mathcal{V}_α$ whose image under the quasisymmetric characteristic is $\mathfrak{S}_α^*$. In this paper, we prove that $\mathcal{V}_α$ admits a distinguished filtration with respect to the basis of Young quasisymmetric Schur functions. This result offers a novel representation-theoretic interpretation of the positive expansion of $\mathfrak{S}_α^*$ in the basis of Young quasisymmetric Schur functions. A key tool in our proof is Mason's analogue of the Robinson-Schensted-Knuth algorithm, for which we establish a version of Green's theorem. As an unexpected byproduct of our investigation, we construct an indecomposable $0$-Hecke module $\mathbf{Y}_α$ whose image under the quasisymmetric characteristic is the Young quasisymmetric Schur function $\hat{\mathscr{S}}_α$. Further properties of this module are also investigated. And, by applying a suitable automorphism twist to this module, we obtain an indecomposable $0$-Hecke module whose image under the quasisymmetric characteristic is the quasisymmetric Schur function $\mathscr{S}_α$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinguished filtrations of the $0$-Hecke modules for dual immaculate quasisymmetric functions Lee, So-Yeon Oh, Young-Tak Representation Theory Combinatorics 20C08, 06A07, 05E10, 05E05 Let $α$ range over the set of compositions. Dual immaculate quasisymmetric functions $\mathfrak{S}_α^*$, introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki, provide a quasisymmetric analogue of Schur functions. They also constructed an indecomposable $0$-Hecke module $\mathcal{V}_α$ whose image under the quasisymmetric characteristic is $\mathfrak{S}_α^*$. In this paper, we prove that $\mathcal{V}_α$ admits a distinguished filtration with respect to the basis of Young quasisymmetric Schur functions. This result offers a novel representation-theoretic interpretation of the positive expansion of $\mathfrak{S}_α^*$ in the basis of Young quasisymmetric Schur functions. A key tool in our proof is Mason's analogue of the Robinson-Schensted-Knuth algorithm, for which we establish a version of Green's theorem. As an unexpected byproduct of our investigation, we construct an indecomposable $0$-Hecke module $\mathbf{Y}_α$ whose image under the quasisymmetric characteristic is the Young quasisymmetric Schur function $\hat{\mathscr{S}}_α$. Further properties of this module are also investigated. And, by applying a suitable automorphism twist to this module, we obtain an indecomposable $0$-Hecke module whose image under the quasisymmetric characteristic is the quasisymmetric Schur function $\mathscr{S}_α$. |
| title | Distinguished filtrations of the $0$-Hecke modules for dual immaculate quasisymmetric functions |
| topic | Representation Theory Combinatorics 20C08, 06A07, 05E10, 05E05 |
| url | https://arxiv.org/abs/2501.11304 |