20.500.12556/RUL-116749 Torsion table for the lie algebra $\mathfrak{nil}_n$ Tabela torzije za Liejevo algebro $\mathfrak{nil}_n$ 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 ▫$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$▫. V članku študiramo Liejev kolobar ▫$\mathfrak{nil}_n$▫ vseh strogo zgornje trikotnih ▫$n\times n$▫ matrik nad ▫$\mathbb{Z}$▫. Za ▫$n\leq 8$▫ smo izračunali celotno homologijo tega kolobarja. Dokazali smo, da v ▫$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$▫ nastopa ▫$p^m$▫-torzija za vse praštevilske potence ▫$p^m\leq n-2$▫. Pri ▫$m=1$▫ je Dwyer pokazal, da je ta meja natančna, tj. ▫$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$▫ ne vsebuje ▫$p$▫-torzije za vse ▫$p>n-2$▫. V splošnem pa za ▫$m>1$▫ meja ni natančna, saj smo pokazali, da ▫$H_\ast(\mathfrak{nil}_8;\mathbb{Z})$▫ vsebuje ▫$8$▫-torzijo. Spotoma smo izpeljali še znano dejstvo, da so rangi prostih delov grup ▫$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$▫ enaki Mahonovim številom (=število permutacij množice ▫$[n]$▫ s ▫$k$▫ inverzijami), preko drugačne izpeljave kot jo je sprva imel Kostant. Na koncu smo določili še algebraično strukturo (kupasti produkti) za ▫$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$▫. algebraic combinatorics algebraic/discrete Morse theory acyclic matching chain complex homological algebra nilpotent Lie algebra torsion table triangular matrices algebraična kombinatorika algebraična/diskretna Morseova teorija aciklično prirejanje verižni kompleks homološka algebra nilpotentna Liejeva algebra tabela torzije trikotne matrike true false false Angleški jezik Slovenski jezik Neznano 2020-06-08 12:25:23 2020-06-08 12:25:27 2022-08-29 03:38:54 0000-00-00 00:00:00 2019 0 0 Str. 3567-3578 no. 9 Vol. 47 2019 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512.81 0092-7872 10.1080/00927872.2019.1567751 18786137 25249792 RAZ_Lampret_Leon_2019.pdf RAZ_Lampret_Leon_2019.pdf 1 4271F3BAE5A7F3E8C299164C936C7405 6a17c35bd8592022da10938f75bca0673123d121e546b797439ecd4a1d078c34 006ef73a-a1b9-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=130424 Fakulteta za matematiko in fiziko 0 0 0