Weak Bruhat interval modules for genomic Schur functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912126857117696 |
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| author | Kim, Young-Hun Yoo, Semin |
| author_facet | Kim, Young-Hun Yoo, Semin |
| contents | Let $λ$ be a partition of a positive integer $n$. The genomic Schur function $U_λ$ was introduced by Pechenik--Yong in the context of the $K$-theory of Grassmannians. Recently, Pechenik provided a positive combinatorial formula for the fundamental quasisymmetric expansion of $U_λ$ in terms of increasing gapless tableaux. In this paper, for each $1 \le m \le n$, we construct an $H_m(0)$-module $\mathbf{G}_{λ;m}$ whose image under the quasisymmetric characteristic is the $m$th degree homogeneous component of $U_λ$ by defining an $H_m(0)$-action on increasing gapless tableaux. We provide a method to assign a permutation to each increasing gapless tableau, and use this assignment to decompose $\mathbf{G}_{λ;m}$ into a direct sum of weak Bruhat interval modules. Furthermore, we determine the projective cover of each summand of the direct sum decomposition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_06575 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weak Bruhat interval modules for genomic Schur functions Kim, Young-Hun Yoo, Semin Representation Theory Combinatorics 20C08, 05E10, 05E05, 14M15 Let $λ$ be a partition of a positive integer $n$. The genomic Schur function $U_λ$ was introduced by Pechenik--Yong in the context of the $K$-theory of Grassmannians. Recently, Pechenik provided a positive combinatorial formula for the fundamental quasisymmetric expansion of $U_λ$ in terms of increasing gapless tableaux. In this paper, for each $1 \le m \le n$, we construct an $H_m(0)$-module $\mathbf{G}_{λ;m}$ whose image under the quasisymmetric characteristic is the $m$th degree homogeneous component of $U_λ$ by defining an $H_m(0)$-action on increasing gapless tableaux. We provide a method to assign a permutation to each increasing gapless tableau, and use this assignment to decompose $\mathbf{G}_{λ;m}$ into a direct sum of weak Bruhat interval modules. Furthermore, we determine the projective cover of each summand of the direct sum decomposition. |
| title | Weak Bruhat interval modules for genomic Schur functions |
| topic | Representation Theory Combinatorics 20C08, 05E10, 05E05, 14M15 |
| url | https://arxiv.org/abs/2211.06575 |