A weak Lehmer code for type $F_4$

Abstract

We provide an algorithm to construct a multicomplex for any lower Bruhat interval of $F_4$, such that its rank--generating function equals that of the Bruhat interval. For Weyl groups, it is always possible to find such a multicomplex thanks to the work of Bj\"{o}rner and Ekedahl. The algorithm is based on only two functions, which weaken the notion of Lehmer code for finite Coxeter groups, motivated by the fact that a strong Lehmer code for type $F_4$ does not exist. We also realize the set of palindromic Poincar\'e polynomials of $F_4$ as an induced subposet of the Bruhat order that forms a lattice.