A weak Lehmer code for type $F_4$

Abstract: We show that, despite the Poincar\'e polynomial of $F_4$ is a product of $q$-analogues, the Bruhat order of $F_4$ does not admit a product of chains as subposet. This answers negatively, in this type, a question by Billey, Fan and Losonczy. In other words, we show that Lehmer codes for type $F_4$ do not exist. Nevertheless, by introducing weak Lehmer codes, we construct explicitly multicomplexes and Lehmer complexes for any lower Bruhat interval in type $F_4$.