Torsion table for the Lie algebra $\frak{nil}_n$
| Autor: | Lampret, Leon, Vavpetič, Aleš |
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| Rok vydání: | 2017 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1080/00927872.2019.1567751 |
| Popis: | We study the Lie ring $\mathfrak{nil}_n$ of all strictly upper-triangular $n\!\times\!n$ matrices with entries in $\mathbb{Z}$. Its complete homology for $n\!\leq\!8$ is computed. We prove that every $p^m$-torsion appears in $H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ for $p^m\!\leq\!n\!-\!2$. For $m\!=\!1$, Dwyer proved that the bound is sharp, i.e. there is no $p$-torsion in $H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ when prime $p\!>\!n\!-\!2$. In general, for $m\!>\!1$ the bound is not sharp, as we show that there is $8$-torsion in $H_\ast(\mathfrak{nil}_8;\mathbb{Z})$. As a sideproduct, we derive the known result, that the ranks of the free part of $H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ are the Mahonian numbers (=number of permutations of $[n]$ with $k$ inversions), using a different approach than Kostant. Furthermore, we determine the algebra structure (cup products) of $H^\ast(\mathfrak{nil}_n;\mathbb{Q})$. Comment: 10 pages, 1 table |
| Databáze: | arXiv |
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