Tight Quotients and Double Quotients in the Bruhat Order

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., “double”) quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are “tight” in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups. 0. Introduction. The Bruhat orderings of Coxeter groups and their parabolic quotients have a long history that originates with the fact that these posets (in the case of finite Weyl groups) record the inclusion of cell closures in generalized flag varieties. Some of the significant early papers on the combinatorial aspects of this subject include the 1977 paper of Deodhar [D1] providing various characterizations of the Bruhat order (including some that will be essential in this work), the 1980 paper of Stanley [St] in which Bruhat orderings of finite Weyl groups and their parabolic quotients are shown to be strongly Sperner, and the 1982 paper of Björner and Wachs in which the Bruhat order is shown to be lexicographically shellable [BW]. *This work was supported by NSF grant DMS–0245385. the electronic journal of combinatorics 11(2) (2005), #R14 1 In this paper, we investigate the explicit assignment of coordinates for the Bruhat order. By a “coordinate assignment” for a poset P , we mean an order-embedding P → R; i.e., an injective map f : P → R such that x < y in P if and only if f(x) < f(y) in the usual (coordinate-wise) partial ordering of R. The minimum such d for which this is possible is known as the order dimension of P , and denoted dim P . For example, Proctor [P1] has given coordinates for the Bruhat orderings of the classical finite Coxeter groups and their quotients, and more recently, Reading [R] has determined the exact order dimensions of the Bruhat orderings of An, Bn, H3, and H4. It would be interesting to have a uniform construction of coordinates for the Bruhat orders of finite Weyl groups, perhaps based directly on the geometry of flag varieties as in Proposition 7.1 of [P1] for type A. For the infinite Coxeter groups, perhaps the most interesting question is the classification of those groups for which the Bruhat ordering is finite-dimensional. Indeed, Reading and Waugh [RW] have shown that there are Coxeter groups whose Bruhat order has infinite order dimension, and infinite Coxeter groups (such as the affine Weyl groups of type A) with finite order dimension. Our initial motivation for this work began with the observation that for each finite Weyl group W and associated affine Weyl group W̃ , the two-sided (parabolic) quotient W\W̃/W may be naturally identified with the dominant part of the co-root lattice. We were surprised to realize that the Bruhat ordering of W\W̃/W is isomorphic to the usual ordering of dominant co-weights: moving up in this Bruhat order is equivalent to adding positive combinations of positive co-roots. (Later, we learned from M. Dyer that this is mentioned explicitly in Section 2 of [L].) This meant that the various remarkable properties of the partial order of dominant (co-)weights (see for example [S2]) could be transfered to the Bruhat ordering of certain two-sided quotients of affine Weyl groups. At this point, we began to investigate more general instances of this phenomenon. Indeed, it is always possible to identify a one-sided parabolic quotient of any Coxeter group with the orbit of a point in the reflection representation, and a two-sided (or “double”) quotient corresponds to the part of an orbit on the positive side of certain hyperplanes. In these terms, a necessary condition for moving up in the Bruhat order requires adding (or subtracting, depending on conventions) positive combinations of positive roots. The interesting question is one of identifying when this necessary condition is sufficient. That is, when do the root coordinates of an orbit, or the portion corresponding to some double quotient, provide an order embedding of the corresponding Bruhat order? The main goal of this paper is to identify these “tight” quotients. An outline of the paper follows. In Section 1, we discuss the details of using the reflection representation of a Coxeter group to model the Bruhat orderings of its parabolic quotients. We also review a key result of Deodhar (see Theorem 1.3) that allows the Bruhat ordering of W to be recovered from its projections onto one-sided or two-sided quotients. In Section 2, we formalize the notion of a tight quotient, and prove a purely ordertheoretic characterization of the tight one-sided quotients (Theorem 2.3): the Bruhat ordering of W/WJ is tight if and only if the Bruhat ordering of WI\W/WJ is a chain for every maximal parabolic subgroup WI of W . We also point out that the Bruhat the electronic journal of combinatorics 11(2) (2005), #R14 2 orderings of minuscule (one-sided) quotients are always tight. In Section 3, we classify the tight one-sided quotients of finite Coxeter groups. We expected the results to include only a few instances beyond the minuscule cases (a frequent outcome in the theory of finite Coxeter groups), but were instead surprised to discover that there are many other examples, including quotients by non-maximal parabolic subgroups. In the course of deriving the classification, we develop two significant necessary conditions for tightness. The first involves the “stratification” of an orbit relative to the action of a parabolic subgroup, and the second involves confining a face of the dominant chamber inside a face of the “double weight arrangement” of hyperplanes (an arrangement that is in general much larger than the usual arrangement defined by the root hyperplanes). In fact, both of these necessary conditions may be used to provide characterizations of tightness (see Lemma 3.3, Theorem 3.9, and Corollary 3.10), although our proofs of the latter two depend a posteriori on the classification. In the final two sections, we focus on the affine Weyl groups. For these groups, there are two natural representations: the first is the usual reflection representation— available for all Coxeter groups—in which the group is represented via linear operators; in the second, one uses affine transformations. In Section 4, we present a dictionary for translating between these two points of view, and prove that there are no one-sided or double quotients that are tight relative to the reflection representation, apart from some trivial cases (Theorem 4.9). In contrast, we show that double quotients with both factors of minuscule type are tight relative to the affine representation (Theorem 4.10). In Section 5, we turn to the classification of quotients of affine Weyl groups that are tight relative to the affine representation. In particular, Theorem 5.12 and Corollary 5.13 provide a classification of all double quotients with a tight embedding in some affine orbit; we find that the left factor must be of minuscule type, but there is a larger number of possibilities for the right factor. The proof has a structure similar to the one in Section 3—we find that there are affine analogues of orbit stratification and the double weight arrangement that provide characterizations of tightness similar to those we develop for finite Coxeter groups (see Theorems 5.10 and 5.11). Acknowledgment. I would like to thank Nathan Reading for many helpful discussions. 1. The Bruhat order. Let (W, S) be a Coxeter system. Via the reflection representation, one may view W as a group of isometries of some real vector space V equipped with a (not necessarily positive definite) inner product 〈 , 〉. In particular, we may associate with W a centrallysymmetric, W -invariant subset Φ ⊂ V − {0} (the root system) so that the reflections in W are the linear transformations sβ : λ 7→ λ − 〈λ, β〉β∨, where β varies over Φ, and β∨ := 2β/〈β, β〉 denotes the co-root corresponding to β. In this framework, the 1 For the details of this construction, we refer the reader to (for example) Chapter 5 of [H], although it should be noted that the normalization 〈β, β〉 = 1 for β ∈ Φ in [H] may be relaxed—rescaling each W -orbit of roots by an arbitrary positive scalar has no significant effect on the general theory. the electronic journal of combinatorics 11(2) (2005), #R14 3 generating set S is the set of simple reflections: for each s ∈ S one may choose a root α (designated to be simple) so that s = sα, and these choices may be arranged so that every root is in either the nonnegative or nonpositive span of the simple roots. Thus Φ is the disjoint union of Φ (the positive roots) and Φ− = −Φ+ (the negative roots). For w ∈ W , let `(w) denote the minimum length of an expression w = s1 · · · sl (si ∈ S). A key relationship between the root system and length is the fact that `(w) < `(sβw) ⇔ w−1β ∈ Φ (w ∈ W, β ∈ Φ), (1.1) and the Bruhat ordering of W may be defined as the transitive closure of the relations