Bruhat Interval Research Articles

Poset Structure Concerning Cylindric Diagrams

Cylindric diagrams admit the structure of infinite $d$-complete posets with natural ordering. The purpose of this paper is to provide a realization of a cylindric diagram as a subset of an affine root system of type A via colored hook lengths and to present several characterizations of its poset structure. Furthermore, the set of order ideals of a cylindric diagram is described as a weak Bruhat interval of the affine Weyl group.

One-skeleton posets of Bruhat interval polytopes

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these polytopes are lattices and classify when the polytopes are simple, thereby resolving open problems and conjectures of Fraser, of Lee–Masuda, and of Lee–Masuda–Park. In particular, we classify when generic torus orbit closures in Schubert varieties are smooth.

Bruhat intervals, subword complexes and brick polyhedra for finite Coxeter groups

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.

On automorphisms of undirected Bruhat graphs

The undirected Bruhat graph \(\Gamma (u,v)\) has the elements of the Bruhat interval [u, v] as vertices, with edges given by multiplication by a reflection. Famously, \(\Gamma (e,v)\) is regular if and only if the Schubert variety \(X_v\) is smooth, and this condition on v is characterized by pattern avoidance. In this work, we classify when \(\Gamma (e,v)\) is vertex-transitive; surprisingly this class of permutations is also characterized by pattern avoidance and sits nicely between the classes of smooth permutations and self-dual permutations. This leads us to a general investigation of automorphisms of \(\Gamma (u,v)\) in the course of which we show that special matchings, which originally appeared in the theory of Kazhdan–Lusztig polynomials, can be characterized, for the symmetric and right-angled groups, as certain \(\Gamma (u,v)\)-automorphisms which are conjecturally sufficient to generate the orbit of e under \({{\,\textrm{Aut}\,}}(\Gamma (e,v))\).

Products of reflections in smooth Bruhat intervals

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.

Classification of Cocovers in the Double Affine Bruhat Order

We classify cocovers of a given element of the double affine Weyl semigroup $W_{\mathcal{T}}$ with respect to the Bruhat order, specifically when $W_{\mathcal{T}}$ is associated to a finite root system that is irreducible and simply laced. We do so by introducing a graphical representation of the length difference set defined by Muthiah and Orr and identifying the cocovering relations with the corners of those graphs. This new method allows us to prove that there are finitely many cocovers of each $x \in W_{\mathcal{T}}$. Further, we show that the Bruhat intervals in the double affine Bruhat order are finite.

Bruhat intervals and parabolic cosets in arbitrary Coxeter groups

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.

Weak Bruhat interval modules of the 0-Hecke algebra

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, restriction, and (anti-)involution twists of weak Bruhat interval modules.

Odd diagrams, Bruhat order, and pattern avoidance

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.Mathematics Subject Classifications: 05A05, 05A15

Singularities of Schubert Varieties within a Right Cell

We describe an algorithm which pattern embeds, in the sense of Woo-Yong, any Bruhat interval of a symmetric group into an interval whose extremes lie in the same right Kazhdan-Lusztig cell. This apparently harmless fact has applications in finding examples of reducible associated varieties of $\mathfrak{sl}_n$-highest weight modules, as well as in the study of $W$-graphs for symmetric groups, and in comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. For example, we are able to systematically produce many negative answers to a question from the 1980s of Borho-Brylinski and Joseph, which had been settled by Williamson via computer calculations only in 2014.

Some combinatorial results on smooth permutations

We show that any smooth permutation $\sigma\in S_n$ is characterized by the set ${\mathbf{C}}(\sigma)$ of transpositions and $3$-cycles in the Bruhat interval $(S_n)_{\leq\sigma}$, and that $\sigma$ is the product (in a certain order) of the transpositions in ${\mathbf{C}}(\sigma)$. We also characterize the image of the map $\sigma\mapsto{\mathbf{C}}(\sigma)$. As an application, we show that $\sigma$ is smooth if and only if the intersection of $(S_n)_{\leq\sigma}$ with every conjugate of a parabolic subgroup of $S_n$ admits a maximum. This also gives another approach for enumerating smooth permutations and subclasses thereof. Finally, we relate covexillary permutations to smooth ones and rephrase the results in terms of the (co)essential set in the sense of Fulton.

Toric Bruhat interval polytopes

For two elements v and w of the symmetric group Sn with v≤w in Bruhat order, the Bruhat interval polytope Qv,w is the convex hull of the points (z(1),…,z(n))∈Rn with v≤z≤w. It is known that the Bruhat interval polytope Qv,w is the moment map image of the Richardson variety Xw−1v−1. We say that Qv,w is toric if the corresponding Richardson variety Xw−1v−1 is a toric variety. We show that when Qv,w is toric, its combinatorial type is determined by the poset structure of the Bruhat interval [v,w] while this is not true unless Qv,w is toric. We are concerned with the problem of when Qv,w is (combinatorially equivalent to) a cube because Qv,w is a cube if and only if Xw−1v−1 is a smooth toric variety. We show that a Bruhat interval polytope Qv,w is a cube if and only if Qv,w is toric and the Bruhat interval [v,w] is a Boolean algebra. We also give several sufficient conditions on v and w for Qv,w to be a cube.

Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

Right-angled Coxeter groups, universal graphs, and Eulerian polynomials

We define a class of representations for any right-angled Coxeter group R and its Hecke algebra H(R). The group R and the algebra H(R) act on any Bruhat interval of any Coxeter system (W,S), once given a suitable function from the set of Coxeter generators of R to the power set of S. The existence of such a function is related to the problem of universality of a graph G2 constructed from the unlabeled Coxeter graph G of (W,S). When G=Pn is a path with n vertices, we conjecture that G2 is n-universal; this property is equivalent to the existence of an action of R and H(R) on the Bruhat intervals of the symmetric group Sn+1, for all right-angled groups with n generators. We prove that (Pn)2 is n-universal for forests. Eulerian polynomials arise as characters of our representations, when the Coxeter graph of R is a path. We also give a formula for the toric h-polynomial of any lower Bruhat interval in a universal Coxeter group Un, using results on the Kazhdan–Lusztig basis of H(Un).

Reduced expressions for the elements in a Bruhat interval

Let (W,S) be a Coxeter system. We study the relation among the reduced expressions of the elements in the interval [y,w]≔{z∈W∣y⩽z⩽w} for any y<w in W. Fix a reduced expression ξ of w. We find some conditions for various expressions ζ to be reduced, where ζ is obtained from ξ by deleting certain factors in S. We investigate the relations among all such ζ which are reduced expressions for either a certain element in [y,w] or some elements with a certain common property.

Flip Posets of Bruhat Intervals

In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph $B(u,v)$ of a Bruhat interval $[u,v]$ into rank posets $P_{i}$ in a way that each $P_{i}$ has a unique maximal chain that is rising under a reflection order. In the case where each $P_{i}$ has rank three, the construction yields a combinatorial description of some terms of the complete $\textbf{cd}$-index as a sum of ordinary $\textbf{cd}$-indices of Eulerian posets obtained from each of the $P_{i}$.

Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

Let (W,S) be a finite Coxeter system with root system R and with set of positive roots R+. For α∈R, v,w∈W, we denote by ∂α, ∂w and ∂w/v the divided difference operators and skew divided difference operators acting on the coinvariant algebra of W. Generalizing the work of Liu [15], we prove that ∂w/v can be written as a polynomial with nonnegative coefficients in ∂α where α∈R+. In fact, we prove the stronger and analogous statement in the Nichols–Woronowicz algebra model for Schubert calculus on W after Bazlov [4]. We draw consequences of this theorem on saturated chains in the Bruhat order, and partially treat the question when ∂w/v can be written as a monomial in ∂α where α∈R+. In an appendix, we study related combinatorics on shuffle elements and Bruhat intervals of length two.

A simple characterization of special matchings in lower Bruhat intervals

We give a simple characterization of special matchings in lower Bruhat intervals (that is, intervals starting from the identity element) of a Coxeter group. As a byproduct, we obtain some results on the action of special matchings.

Special matchings in Coxeter groups

Special matchings are purely combinatorial objects associated with a partially ordered set, which have applications in Coxeter group theory. We provide an explicit characterization and a complete classification of all special matchings of any lower Bruhat interval. The results hold in any arbitrary Coxeter group and have also applications in the study of the corresponding parabolic Kazhdan–Lusztig polynomials.

The generalized lifting property of Bruhat intervals

In Tsukerman and Williams (Adv Math 285: 766–810, 2015), it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this paper, we show that a Coxeter group satisfies this property if and only if it is finite and simply-laced.