We study the appearance of notable interval structures -- lattices, modular lattices, distributive lattices, and boolean lattices -- in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for principal order ideals, including pattern characterizations and enumerations for the symmetric group. This segues naturally into a similar analysis for arbitrary intervals, although the results are less characterizing for the Bruhat order at this generality. In counterpoint, however, we obtain a full characterization for intervals starting at rank one in the symmetric group, for each of the four structure types, in each of the two posets. Each category can be enumerated, with intriguing connections to Fibonacci and Catalan numbers. We conclude with suggestions for further directions and questions, including an interesting analysis of the intervals formed between a permutation and each generator in its support.
We study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root configurations for subword complexes, culminating in the notion of brick polyhedra for general subword complexes.
The number of Bruhat intervals in Coxeter groups is finite, and for the first few lengths, the intervals were described up to an isomorphism by A. Hultman using the correspondence between Bruhat intervals and cell decompositions of a 2d sphere and straightforward computations. The main purpose of this paper consists of a description of the intervals in higher dimensions, as the Hultman's geometric method is hard to apply due to rapidly growing with length number of nonisomorphic intervals. We construct an invariant on subintervals in the Bruhat graphs, using their specific properties. This giv…
In this paper, we prove that if the dual of a Bruhat interval in a Weyl group is a zircon, then that interval is rationally smooth. Investigating when the converse holds, and drawing inspiration from conjectures by Delanoy, leads us to pose two conjectures. If true, they imply that for Bruhat intervals in type $A$, duals of smooth intervals, zircons, and being isomorphic to lower intervals are all equivalent. As a verification, we have checked our conjectures in types $A_n$, $n\leq 8$.
In [Journal of Pure and Applied Algebra {224} (2020), no 12, 106449], V. Mazorchuk and R. Mrđen (with some help by A. Hultman) prove that, given a Weyl group, the intersection of a Bruhat interval with a parabolic coset has a unique maximal element and a unique minimal element. We show that such intersections are actually Bruhat intervals also in the case of an arbitrary Coxeter group.
In this paper, we study the decomposition of Bruhat intervals in a Coxeter group with respect to cosets of a parabolic subgroup. Our main result is that the intersection of a lower Bruhat interval with a parabolic coset contains a unique maximal element. As an application, we give a decomposition formula for the Poincaré polynomial of a Coxeter group element. We also show that the fibers of standard parabolic projection maps on Schubert varieties are themselves Schubert varieties.
A permutation is called smooth if the corresponding Schubert variety is smooth. Gilboa and Lapid prove that in the symmetric group, multiplying the reflections below a smooth element $w$ in Bruhat order in a compatible order yields back the element $w$. We strengthen this result by showing that such a product in fact determines a saturated chain $e \to w$ in Bruhat order, and that this property characterizes smooth elements.
Westwood High School（韦斯特伍德高中）；Cranbrook Kingswood Upper School（克兰布鲁克金伍兹高中部）；Oklahoma State University（俄克拉荷马州立大学）；Texas State University（德克萨斯州立大学）；West Windsor-Plainsboro High School North（西温莎-普莱恩斯伯勒北高中）；Seven Lakes High School（七湖高中）
Björner-Ekedahl 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 and of Lakshmibai-Sandhya 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 c…
The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restri…
Let $W$ be a finite reflection group. For a given $w \in W$, the following assertion may or may not be satisfied: (*) The principal Bruhat order ideal of $w$ contains as many elements as there are regions in the inversion hyperplane arrangement of $w$. We present a type independent combinatorial criterion which characterises the elements $w\in W$ that satisfy (*). A couple of immediate consequences are derived: (1) The criterion only involves the order ideal of $w$ as an abstract poset. In this sense, (*) is a poset-theoretic property. (2) For $W$ of type $A$, another characterisation of (*), …
We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id,pi] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations pi such that [id,pi] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincare polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincare polynomial of some particularly interesting intervals in the finite Weyl grou…
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究所）
We study the partial orders induced on Wachs and signed Wachs permutations by the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their ordering and covering relations, and compute their characteristic polynomials, when partially ordered by Bruhat order, and determine their structure explicitly when partially ordered by right weak order.
The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.
For each permutation $w$, we can construct a collection of hyperplanes $\mathcal{A}_w$ according to the inversions of $w$, which is called the inversion hyperplane arrangement associated to $w$. It was conjectured by Postnikov and confirmed by Hultman, Linusson, Shareshian and Sjöstrand that the number of regions of $\mathcal{A}_w$ is less than or equal to the number of permutations below $w$ in the Bruhat order, with the equality holds if and only if $w$ avoids the four patterns 4231, 35142, 42513 and 351624. In this paper, we show that the number of regions of $\mathcal{A}_w$ is greater than…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Universidad de Chile（智利大学）；Universidad de Talca（塔尔卡大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）；University of Sydney（悉尼大学）
