On Lusztig-Dupont homology of flag complexes

TL;DR

This paper investigates Lusztig-Dupont homology of flag complexes associated with finite vector spaces, providing explicit bases and minimal support size results for twisted homology modules, advancing understanding of their algebraic and combinatorial properties.

Contribution

It constructs an explicit basis for the homology module D^1(V) and establishes minimal support size bounds for cycles in twisted homology, extending previous results.

Findings

Explicit basis for D^1(V) constructed.

Minimal support size of cycles in twisted homology determined.

Extension of results by Smith and Yoshiara to broader cases.

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field of order $q$. The spherical building $X_V$ associated with $GL(V)$ is the order complex of the nontrivial linear subspaces of $V$. Let $\mathfrak{g}$ be the local coefficient system on $X_V$, whose value on the simplex $\sigma=[V_0 \subset \cdots \subset V_p] \in X_V$ is given by $\mathfrak{g}(\sigma)=V_0$. Following the work of Lusztig and Dupont, we study the homology module $D^k(V)=\tilde{H}_{n-k-1}(X_V;\mathfrak{g})$. Our results include a construction of an explicit basis of $D^1(V)$, and the following twisted analogue of a result of Smith and Yoshiara: For any $1 \leq k \leq n-1$, the minimal support size of a non-zero $(n-k-1)$-cycle in the twisted homology $\tilde{H}_{n-k-1}(X_V;\wedge^k \mathfrak{g})$ is $\frac{(n-k+2)!}{2}$.