8 research outputs found
A Suspension Lemma for Bounded Posets
Let and be bounded posets. In this note, a lemma is introduced that
provides a set of sufficient conditions for the proper part of being
homotopy equivalent to the suspension of the proper part of~. An application
of this lemma is a unified proof of the sphericity of the higher Bruhat orders
under both inclusion order (a known proved earlier by Ziegler) and single step
inclusion order (which was not previously known)
Properties of four partial orders on standard Young tableaux
Let SYT_n be the set of all standard Young tableaux with n cells. After
recalling the definitions of four partial orders, the weak, KL, geometric and
chain orders on SYT_n and some of their crucial properties, we prove three main
results: (i)Intervals in any of these four orders essentially describe the
product in a Hopf algebra of tableaux defined by Poirier and Reutenauer. (ii)
The map sending a tableau to its descent set induces a homotopy equivalence of
the proper parts of all of these orders on tableaux with that of the Boolean
algebra 2^{[n-1]}. In particular, the M\"obius function of these orders on
tableaux is (-1)^{n-3}. (iii) For two of the four orders, one can define a more
general order on skew tableaux having fixed inner boundary, and similarly
analyze their homotopy type and M\"obius function.Comment: 24 pages, 3 figure
Gallery Posets of Supersolvable Arrangements
We introduce a poset structure on the reduced galleries in a supersolvable arrangement of hyperplanes. In particular, for Coxeter groups of type A or B, we construct a poset of reduced words for the longest element whose Hasse diagram is the graph of reduced words. Using Rambau's Suspension Lemma, we show that these posets are homotopy equivalent to spheres. We furthermore conjecture that its intervals are either homotopy equivalent to spheres or are contractible. One may view this as a analogue of a result of Edelman and Walker on the homotopy type of intervals of a poset of chambers of a hyperplane arrangement