Self-dual intervals in the Bruhat order
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Self-dual intervals in the Bruhat order Christian Gaetz1 · Yibo Gao1 Accepted: 22 October 2020 © Springer Nature Switzerland AG 2020
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 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. Mathematics Subject Classification 05E99 · 14M15
1 Introduction We say a complex projective variety X has a cellular decomposition if X is covered by the disjoint open sets {Ci }, each isomorphic to affine space of some dimension, and such that each boundary C j \C j is a union of some of the {Ci }. Given a variety with such a decomposition, it is natural, following Stanley [14], to define a partial order Q X on the {Ci } by setting Ci ≤ C j whenever Ci ⊆ C j . When X = G/B, the quotient of a complex semisimple algebraic group by a Borel subgroup, the Bruhat decomposition
C.G. was partially supported by an NSF Graduate Research Fellowship under Grant No. 1122374.
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Christian Gaetz [email protected] Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA 0123456789().: V,-vol
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C. Gaetz, Y. Gao
G=
Bw B
w∈W
induces a cellular decomposition {Bw B/B |w ∈ W } of X , where W is the Weyl group of G. In this case the partial order Q X on W is the well known Bruhat order. For w ∈ W the closure X w = Bw B/B itself has the cellular decomposition {Bu B/B|u ∈ W , u ≤ w}, and so its poset of cells Q X w is the interval [e, w] in Bruhat order on W below the element w. The varieties X w are called Schubert varieties. Much of the structure of the Bruhat order is well-understood combinatorially; see Sect. 2 for some basic definitions and results. It is graded with the rank of an element w being the length (w) in the Weyl group, it has minimal element e, the identity element of W and maximal element w0 , the longest element of W . A great deal of work has been done on the structure of intervals [e, w] in Bruhat order [3,6,15]. Most of this paper will focus on the “type An−1 ” case, where the Weyl group W is the symmetric group Sn . Fo
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