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 \(X_w\) 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.

Björner和Ekedahl（Ann Math（2）170（2）：799–817，2009）证明，以Bruhat顺序排列的一般区间[ e，  w ]是“重头”的，至少第i个元素的个数排名第i位。卡雷尔的著名结果（在：代数群及其概括：经典方法（大学公园，宾夕法尼亚州，1991年），纯数学研讨会论文集，第56卷，第53-61页。美国数学协会，普罗维登斯，RI，1994年）和Lakshmibai和Sandhya（1990年印度科学学会科学学报100（1）：45–52，Proc）给出等式：[ e，  w ]仅当置换w时才是秩对称的。避免使用模式3412和4231，而这些恰好是w，从而舒伯特变量\（X_w \）是平滑的。在本文中，我们研究了秩对称区间[ e，  w ]的更精细结构，超越了它们的秩函数。尤其是，我们表明，如果一个计数涵盖了不同等级之间的关系，则这些时间间隔仍然是“头重脚轻”的事情。当[ e，w ]作为对偶作为自对偶时， 会发生这种情况下的相等情况。我们通过模式避免和其他几种方式来表征这些w。