Homology of Segre powers of Boolean and subspace lattices
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908704548323328 |
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| author | Li, Yifei Sundaram, Sheila |
| author_facet | Li, Yifei Sundaram, Sheila |
| contents | Segre products of posets were defined by Björner and Welker (2005). We investigate the homology representations of the $t$-fold Segre power $B_n^{(t)}$ of the Boolean lattice $B_n$. The direct product $\sym_n^{\times t}$ of the symmetric group $\sym_n$ acts on the homology of rank-selected subposets of $B_n^{(t)}$. We give an explicit formula for the decomposition into $\sym_n^{\times t}$-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group $\sym_n$. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the $\sym_n^{\times t}$-module coincides with the corresponding rank-selected invariant of the $t$-fold Segre power of the subspace lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homology of Segre powers of Boolean and subspace lattices Li, Yifei Sundaram, Sheila Combinatorics Representation Theory 05E10, 05E05, 20C30, 55P99 Segre products of posets were defined by Björner and Welker (2005). We investigate the homology representations of the $t$-fold Segre power $B_n^{(t)}$ of the Boolean lattice $B_n$. The direct product $\sym_n^{\times t}$ of the symmetric group $\sym_n$ acts on the homology of rank-selected subposets of $B_n^{(t)}$. We give an explicit formula for the decomposition into $\sym_n^{\times t}$-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group $\sym_n$. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the $\sym_n^{\times t}$-module coincides with the corresponding rank-selected invariant of the $t$-fold Segre power of the subspace lattice. |
| title | Homology of Segre powers of Boolean and subspace lattices |
| topic | Combinatorics Representation Theory 05E10, 05E05, 20C30, 55P99 |
| url | https://arxiv.org/abs/2408.08421 |