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Torsion table for the lie algebra $\mathfrak{nil}_n$
Lampret, Leon (Author), Vavpetič, Aleš (Author)

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Abstract
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 ▫$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$▫.

Language:English
Keywords:algebraic combinatorics, algebraic/discrete Morse theory, acyclic matching, chain complex, homological algebra, nilpotent Lie algebra, torsion table, triangular matrices
Tipology:1.01 - Original Scientific Article
Organization:FMF - Faculty of Mathematics and Physics
Year:2019
Number of pages:str. 3567-3578
Numbering:Vol. 47, no. 9
UDC:512.81
ISSN on article:0092-7872
DOI:10.1080/00927872.2019.1567751 Link is opened in a new window
COBISS.SI-ID:18786137 Link is opened in a new window
Views:122
Downloads:66
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Record is a part of a journal

Title:Communications in algebra
Shortened title:Comm. algebra
Publisher:M. Dekker.
ISSN:0092-7872
COBISS.SI-ID:25249792 This link opens in a new window

Secondary language

Language:Slovenian
Title:Tabela torzije za Liejevo algebro $\mathfrak{nil}_n$
Abstract:
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 ▫$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$▫.

Keywords:algebraična kombinatorika, algebraična/diskretna Morseova teorija, aciklično prirejanje, verižni kompleks, homološka algebra, nilpotentna Liejeva algebra, tabela torzije, trikotne matrike

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