An improved tableau criterion for Bruhat order

Abstract

To decide whether two permutations are comparable in Bruhat order of S n with the well-known tableau criterion requires (n2) comparisons of entries in certain sorted arrays. We show that to decide whether x ≤ y only d1 + d2 + ... + dk of these comparisons are needed, where {d1, d2, ..., dk} - {i|x(i) > x(i + 1)}. This is obtained as a consequence of a sharper version of Deodhar's criterion, which is valid for all Coxeter groups.