Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8903746 | Journal of Combinatorial Theory, Series A | 2018 | 21 Pages |
Abstract
We prove a dual version of Ãystein Ore's theorem on distributive intervals in the subgroup lattice of finite groups, having a nonzero dual Euler totient ÏË. For any Boolean group-complemented interval, we observe that ÏË=Ïâ 0 by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that ÏË 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 PË=CË(H,G) is Cohen-Macaulay and the nontrivial reduced Betti number of the order complex Î(P) is ÏË, 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Mamta Balodi, Sebastien Palcoux,