2018/07/13 by Meshulam, Roy, Zerbib, Shira
#20E42 #55U10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.05297
Let V be an n-dimensional vector space over the finite field of order q. The spherical building XV associated with GL(V) is the order complex of the nontrivial linear subspaces of V. Let \mathfrakg be the local coefficient system on XV, whose value on the simplex σ=[V0 ⊂ ⋯ ⊂ Vp] ∈ XV is given by \mathfrakg(σ)=V0. Following the work of Lusztig and Dupont, we study the homology module Dk(V)=Hn-k-1(XV;\mathfrakg). Our results include a construction of an explicit basis of D1(V), and the following twisted analogue of a result of Smith and Yoshiara: For any 1 ≤ k ≤ n-1, the minimal support size of a non-zero (n-k-1)-cycle in the twisted homology Hn-k-1(XV;\wedgek \mathfrakg) is ((n-k+2)!)/(2).