2016/04/30 by Mamta Balodi, Sebastien Palcoux · 1 citation
Mathematics · #math.GR #math.CO #math.OA #math.QA #math.RT #msc:06A11 #msc:05E15 #msc:06C15 #msc:05E10 #msc:05E45 #msc:16E65
paper · pdf · doi:10.1016/j.jcta.2018.02.004
published as Journal of Combinatorial Theory, Series A, 157 (2018), 49-69 · 16 pages; shortened version
arxiv created 2018/02/10 · arxiv updated 2018/02/28
We prove a dual version of Øystein Ore's theorem on distributive intervals in the subgroup lattice of finite groups, having a nonzero dual Euler totient φ. For any Boolean group-complemented interval, we observe that φ = φ≠ 0 by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that φ is always nonzero for Boolean intervals. In order to investigate it, we prove that for any Boolean group-complemented interval [H,G], the graded coset poset P = C(H,G) is Cohen-Macaulay and the nontrivial reduced Betti number of the order complex Δ(P) is φ, so nonzero. We deduce that these results are true beyond the group-complemented case with |G:H|<32. One observes that they are also true when H is a Borel subgroup of G.