Self-dual intervals in the Bruhat order

Abstract

Björner and Ekedahl (Ann Math (2) 170(2):799–817, 2009) prove that general intervals [e, w] in Bruhat order are “top-heavy”, with at least as many elements in the i-th corank as the i-th rank. Well-known results of Carrell (in: Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), volume 56 of proceedings of symposium on pure mathematics, pp 53–61. American Mathematical Society, Providence, RI, 1994) and of Lakshmibai and Sandhya (Proc Indian Acad Sci Math Sci 100(1):45–52, 1990) give the equality case: [e, w] is rank-symmetric if and only if the permutation w avoids the patterns 3412 and 4231 and these are exactly those w such that the Schubert variety X[subscript w] is smooth. In this paper we study the finer structure of rank-symmetric intervals [e, w], beyond their rank functions. In particular, we show that these intervals are still “top-heavy” if one counts cover relations between different ranks. The equality case in this setting occurs when [e, w] is self-dual as a poset; we characterize these w by pattern avoidance and in several other ways.