Torsion table for the lie algebra $\mathfrak{nil}_n$Lampret, Leon (Avtor)
Vavpetič, Aleš (Avtor)
algebraic combinatoricsalgebraic/discrete Morse theoryacyclic matchingchain complexhomological algebranilpotent Lie algebratorsion tabletriangular matricesWe 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 ▫$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$▫.20192020-06-08 12:25:23Neznano116749sl