Leon Lampret (Author), Aleš Vavpetič (Author)

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;algebraic combinatorics;algebraic/discrete Morse theory;acyclic matching;chain complex;homological algebra;nilpotent Lie algebra;torsion table;triangular matrices;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FMF - Faculty of Mathematics and Physics
UDC: 512.81
COBISS: 18786137 Link will open in a new window
ISSN: 0092-7872
Views: 478
Downloads: 225
Average score: 0 (0 votes)
Metadata: JSON JSON-RDF JSON-LD TURTLE N-TRIPLES XML RDFA MICRODATA DC-XML DC-RDF RDF

Other data

Secondary language: Slovenian
Secondary title: Tabela torzije za Liejevo algebro nil[spodaj]n
Secondary 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 ▫$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$▫.
Secondary keywords: algebraična kombinatorika;algebraična/diskretna Morseova teorija;aciklično prirejanje;verižni kompleks;homološka algebra;nilpotentna Liejeva algebra;tabela torzije;trikotne matrike;
Pages: str. 3567-3578
Volume: ǂVol. ǂ47
Issue: ǂno. ǂ9
Chronology: 2019
DOI: 10.1080/00927872.2019.1567751
ID: 11807750