Self-dual intervals in the Bruhat order

Abstract

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 proceed-ings of symposium on pure mathematics, pp 53–61. American Mathematical Society, Providence, RI, 1994) and of Lakshmibai and Sandhya (Proc Indian Acad Sci MathSci 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] 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.