SCHUBERT VARIETIES AND FREE BRAIDEDNESS

R.M. Green and J. Losonczy

Department of Mathematics

University of Colorado Campus Box 395 Boulder, CO 80309-0395 USA

E-mail: rmg@euclid.colorado.edu

Department of Mathematics Long Island University 720 Northern Boulevard Brookville, NY 11548-1319

USA E-mail: losonczy@liu.edu

Abstract. We give a simple necessary and sufficient condition for a Schubert variety X_{w} to be smooth when w is a freely braided element of a simply laced Weyl group; such elements were introduced by the authors in a previous work. This generalizes in one direction a result of Fan concerning varieties indexed by short-braid avoiding elements. We also derive generating functions for the freely braided elements that index smooth Schubert varieties. All results are stated and proved only for the simply laced case.

To appear in Transformation Groups

Introduction

The “freely braided elements” of a simply laced Coxeter group were defined in

[7]. The idea behind the definition is that although it may be necessary to use long

braid relations in order to pass between two reduced expressions for a freely braided

Typeset by AMS-TEX 2 R.M. GREEN AND J. LOSONCZY

1991 Mathematics Subject Classification. 20F55, 14M15.

element, the necessary long braid relations in a certain sense do not interfere with one another.

In this paper, we study Schubert varieties indexed by freely braided elements of a simply laced Weyl group; the non-simply-laced case will not be considered. Specifically, we are interested in the problem of determining which of these varieties are smooth. The general problem of characterising smoothness for Schubert varieties has been investigated by numerous authors in recent years. For example, Lakshmibai–Sandhya [12] have shown that in the type A_{n} setting, a Schubert variety X_{w} is nonsingular if and only if the permutation w avoids the patterns 3412 and 4231. Billey [1] has obtained similar pattern-avoidance conditions for rational smoothness in types B_{n}, C_{n} and D_{n}. General necessary and sufficient conditions for smoothness have been given by Kumar [11], and recently Billey–Postnikov [3] have found a criterion for smoothness in terms of patterns in root systems. Important results have also been obtained by Carrell, Peterson and others; for an overview of the subject, the reader is referred to [2].

A main source of motivation for the present work is the paper [6], in which Fan studies short-braid avoiding elements of an arbitrary Weyl group and determines which of these elements correspond to smooth Schubert varieties. Fan found that for a short-braid avoiding element w, the corresponding Schubert variety X_{w} is nonsingular if and only if w is a product of ℓ(w) distinct generators. Formulae for the enumeration of such elements were also provided. Since in the simply laced setting short-braid avoiding elements are a special case of freely braided elements, it is natural to try to generalize Fan’s results to the freely braided case, and this is a principal goal of the present article. Theorem 3.2 provides a particularly simple characterisation of those freely braided w for which X_{w} is smooth. A by-product of our argument is that we can describe exactly when the deletion of a letter from a reduced expression for a freely braided element results in another (not necessarily freely braided) reduced expression (see Proposition 2.4 and Remark 2.5). The

SCHUBERT VARIETIES AND FREE BRAIDEDNESS 3

answer here is perhaps the simplest that can be expected, and generalizes in one direction a deletion result of Fan [6, Theorem 1] concerning short-braid avoiding elements of Weyl groups. Recently, Hagiwara–Ishikawa–Tagawa [9] have generalized the same deletion result in a different way to a wider class of Coxeter groups.

In Theorem 4.2, we present generating functions for the number of smooth varieties indexed by freely braided elements. These may easily be converted to explicit formulae using standard methods.

1. Preliminaries

Let W be a simply laced Coxeter group with set of distinguished generators S = {s_{i} : i ∈ I} and Coxeter matrix (m_{ij})_{i,j∈I}. Denote by I^{∗} the free monoid on I, and let ϕ : I^{∗} ⟶ W be the surjective morphism of monoid structures satisfying ϕ(i) = s_{i} for all i ∈ I. We say that a word i ∈ I^{∗} represents its image w = ϕ(i) ∈ W; further, if the length of i (i.e., the number of factors used to express i as a product of letters from I) is minimal among the lengths of all the words that represent w, then we call i a reduced expression for w. The length of w, denoted by ℓ(w), is then equal to the length of i.

Let V be a real vector space with basis {α_{i} : i ∈ I}, and denote by B the Coxeter form on V associated to W. This is the symmetric bilinear form satisfying B(α_{i}, α_{j}) = − cos \frac{π}{m_{ij}} for all i, j ∈ I. We view V as the underlying space of a reflection representation of W, determined by the equalities s_{i}α_{j} = α_{j} −2B(α_{j}, α_{i})α_{i} for all i, j ∈ I.

Denote by Φ the root system of W, i.e., {wα_{i} : w ∈ W and i ∈ I}. Let Φ^{+} be the set of all β ∈ Φ such that β is expressible as a linear combination of the α_{i} with nonnegative coefficients, and let Φ^{−} = −Φ^{+}. We have Φ = Φ^{+} ∪Φ^{−} (disjoint). The elements of Φ^{+} (respectively, Φ^{−}) are called positive (respectively, negative) roots. The α_{i} are also referred to as simple roots.

Associated to each w ∈ W is the inversion set Φ(w) = Φ^{+} ∩ w^{−1}(Φ^{−}), which

4 R.M. GREEN AND J. LOSONCZY

has cardinality ℓ(w). Given any reduced expression i_{1}i_{2} ⋯ i_{n} for w, we have Φ(w) = {r_{1}, r_{2}, …, r_{n}}, where r_{1} = α_{i_{n}} and r_{l} = s_{i_{n}} ⋯ s_{i_{n−l+2}}(α_{i_{n−l+1}}) for all l ∈ {2, …, n}. The sequence r̄ = (r_{1}, r_{2}, …, r_{n}) is called the root sequence of i_{1}i_{2} ⋯ i_{n}, or a root sequence for w. Note that any initial segment of a root sequence is also a root sequence for some element of W.

Let i, j ∈ I^{∗} and let i, j, k ∈ I. We call the substitution iijj → ijij a commutation or short braid move if m_{ij} = 2, and if m_{ij} = 3, we call iijij → ijijj a long braid move. Applying a braid move to a reduced expression corresponds to applying a permutation to the root sequence of that reduced expression [7, Proposition 3.1.1]. Specifically, let w ∈ W and suppose that iijj is a reduced expression for w. Let r̄ = (r_{l}) be the associated root sequence, and let n be the length of j. Then m_{ij} = 2 if and only if r_{n+1} and r_{n+2} are mutually orthogonal, in which case the root sequence r̄′ of ijij can be obtained from r̄ by interchanging r_{n+1} and r_{n+2}. Employing again the terminology used above for words, we say that the passage from r̄ to r̄′ is obtained by a commutation or short braid move. Suppose now that iijkj is a reduced expression for w, and let r̄ = (r_{l}) be the associated root sequence. Again, denote the length of j by n. Then i = k if and only if r_{n+1} + r_{n+3} = r_{n+2}, in which case the root sequence r̄′ of ijijj can be obtained from r̄ by interchanging r_{n+1} and r_{n+3}. In this instance, we say that the passage from r̄ to r̄′ is obtained by a long braid move.

Let w ∈ W. Any subset of Φ(w) of the form {α, β, α + β} is called an inversion triple of w. We say that an inversion triple T of w is contractible if there is a root sequence for w in which the elements of T appear consecutively (in some order). The number of contractible inversion triples of w is denoted by N(w). If the contractible inversion triples of w are pairwise disjoint, then w is said to be freely braided.

Remark 1.1. Suppose that W is of type A_{n}. Then the freely braided elements w ∈ W can be characterised by a pattern avoidance condition involving four patterns (this was pointed out by one of the referees, and is also discussed in [7, §5.1]). More precisely, a permutation w is freely braided if and only if its 1-line notation avoids 3421, 4231, 4312 and 4321. This fact will not be needed in the sequel.

2. A deletion property of freely braided elements

We introduce a partial order ≤ on Φ by writing α ≤ β if β − α is a nonnegative linear combination of simple roots. The following lemma will be used repeatedly in the proof of Proposition 2.2.

Lemma 2.1. Let w ∈ W and let r̄ be a root sequence for w. Suppose that r̄ = (…, α, β, …), where α is not orthogonal to β relative to the Coxeter form, and α ≰ β. Then {α − β, α, β} is a contractible inversion triple of w.

Proof. Let ijij be the reduced expression corresponding to r̄, parsed in such a way that α = ϕ(j)^{−1}(α_{i}) and β = ϕ(j)^{−1}s_{i}(α_{j}). We first prove that {α − β, α, β} is an inversion triple of w by showing that ϕ(j) has a reduced expression starting with j. For this, it suffices by [10, Proposition 5.7, Theorem 5.8] to verify that ϕ(j)^{−1}(α_{j}) lies in Φ^{−}.

Since the root α is not orthogonal to β, we have m_{ij} = 3, and hence s_{i}(α_{j}) = α_{j} + α_{i}. We now compute ϕ(j)^{−1}(α_{j}) = ϕ(j)^{−1}(α_{j} + α_{i} − α_{i}) = ϕ(j)^{−1}s_{i}(α_{j}) − ϕ(j)^{−1}(α_{i}) = β − α, which must lie in Φ^{−} since α ≰ β.

Thus, {α − β, α, β} is an inversion triple of w. Since α and β are adjacent in r̄, [7, Proposition 3.2.1] now implies that {α − β, α, β} is contractible. □

We define the height of any root β to be the sum of the coefficients used to express β as a linear combination of the simple roots.

Proposition 2.2. Let w ∈ W be freely braided and suppose that α is the highest root of an inversion triple of w. Then α is the highest root of a contractible inversion triple of w.

Note. In the type A_{n} setting, every inversion triple is contractible, but this is not true in general (see [7, Example 2.2.3, Proposition 5.1.1]).

6 R.M. GREEN AND J. LOSONCZY

Proof. Assume the contrary. Then, by Lemma 2.1, in every root sequence for w, the root α cannot be directly to the left of a root belonging to an inversion triple of w in which α is the highest root. On the other hand, by hypothesis, α is the highest root of some inversion triple of w. Choose a root sequence (…, α, β_{1}, …, β_{n}, γ, …) for w with the property that {α − γ, α, γ} is an inversion triple of w, and n ≥ 1 is as small as possible (over all inversion triples of w having α as highest root).

Now, β_{n} is not orthogonal to γ by the minimality of n. We claim that β_{n} ≤ γ. Assume otherwise. Then, according to Lemma 2.1, {β_{n} − γ, β_{n}, γ} is a contractible inversion triple of w. By [7, Lemma 4.2.2], we can commute β_{n} − γ to the right, if necessary, so that it is adjacent to β_{n}. We cannot have β_{n} −γ = α, or else, after the commutations, our sequence would take the form (…, β_{1}, …, β_{n−1}, α, β_{n}, γ, …); a single long braid move would then give us (…, β_{1}, …, β_{n−1}, γ, β_{n}, α, …), with α lying to the right of both γ and α − γ, an impossibility [7, Remark 2.2.2]. So β_{n} − γ ≠ α. But then our sequence (after the commutations) looks like (…, α, β_{1}, …, β_{n−1}, β_{n}−γ, β_{n}, γ, …), where β_{n}−γ may or may not equal one of the β_{l}. Either way, performing a long braid move gives (…, α, β_{1}, …, β_{n−1}, γ, β_{n}, β_{n} − γ, …), contradicting the minimality of n. The claim is established, i.e., β_{n} ≤ γ.

Let γ^{∗} be a ≤-minimal element in {β_{l} : β_{l} ≤ γ}. We claim that there is an index m such that β_{m} is to the left of γ^{∗} and β_{m} ⊥̸ γ^{∗}. If such an m did not exist, then we could commute γ^{∗} to the left to obtain the sequence (…, α, γ^{∗}, …, γ, …). The root γ^{∗} is not orthogonal to α, because if it were, then we could commute it past α and contradict the minimality of n. We also have α ≰ γ^{∗} by the definition of γ^{∗}. Hence, by Lemma 2.1, {α − γ^{∗}, α, γ^{∗}} is a contractible triple, contradicting the fact that α is not the highest root of any contractible triple of w. So there does indeed exist such an m, and we may assume that β_{m} is directly to the left of γ^{∗} in our chosen root sequence (…, α, β_{1}, …, β_{n}, γ, …).

We have β_{m} ≰ γ^{∗} by our choice of γ^{∗}. By Lemma 2.1 again, {β_{m} − γ^{∗}, β_{m}, γ^{∗}} is a contractible triple of w. If necessary, we can commute β_{m} − γ^{∗} to the right in

SCHUBERT VARIETIES AND FREE BRAIDEDNESS 7

our chosen sequence so that it is adjacent to β_{m} (by [7, Lemma 4.2.2]). We can then apply a long braid move so that the three roots of our triple appear in the order γ^{∗}, β_{m}, β_{m} − γ^{∗}. If β_{m} − γ^{∗} were to equal α, then this long braid move would place α closer to γ than before, a contradiction. So β_{m} − γ^{∗} ≠ α.

Suppose that β_{m}−γ^{∗} was to the left of α before the commutations and long braid move of the previous paragraph. Since we commuted β_{m} − γ^{∗} past α, β_{1}, …, β_{m−1} in order to put it next to β_{m}, it is orthogonal to all of these vectors. We claim that the root γ^{∗} is also orthogonal to α, β_{1}, …, β_{m−1}. To see this, assume otherwise. Then, after the long braid move of the previous paragraph, we can commute γ^{∗} to the left until it is adjacent to a root δ among α, β_{1}, …, β_{m−1} with which it is not orthogonal. Since γ^{∗} cannot be ≥ δ, Lemma 2.1 implies that γ^{∗} and δ belong to the same contractible triple of w. This is a contradiction, since γ^{∗} cannot belong to any contractible triple of w other than {β_{m} − γ^{∗}, β_{m}, γ^{∗}}, owing to the fact that w is freely braided. The claim is established. It follows that each root in {β_{m} − γ^{∗}, β_{m}, γ^{∗}} is orthogonal to α, β_{1}, …, β_{m−1}, and so can be commuted to the left past all of them, contradicting the minimality of n.

Suppose finally that β_{m} − γ^{∗} is one of the β_{l}. Even here, after applying the long braid move mentioned two paragraphs above, we can commute γ^{∗} to the left past α (by the argument of the previous paragraph), contradicting the minimality of n. This last contradiction exhausts all possibilities for β_{m} − γ^{∗}. □

Let i be a word in I^{∗} and suppose that i can be written as u_{0}b_{1}u_{1}b_{2}u_{2} ⋯ b_{p}u_{p}, where each b_{l} is of the form iji for some i, j ∈ I with m_{ij} = 3. Then we call b_{1}, b_{2}, …, b_{p} a braid sequence for i. If i ∈ I^{∗} is reduced and w = ϕ(i) is freely braided, then we say that i is contracted provided there exists a braid sequence for i with p = N(w) terms. Proposition 2.3 [8, Proposition 3.1.3]. Let w ∈ W be freely braided.

(i) There exists a contracted reduced expression for w. (ii) Any contracted reduced expression for w has a unique braid sequence with N(w)

8 R.M. GREEN AND J. LOSONCZY

terms. □

The proof of the next proposition adapts an argument of Fan [6, Theorem 1]. We continue to assume that W is simply laced.

Proposition 2.4. Assume that W is finite. Let w be a freely braided element of W, and let i = i_{1} ⋯ i_{n} be a contracted reduced expression for w with braid sequence b_{1}, …, b_{N(w)}. Then the expression obtained from i by deleting any letter, except a middle letter from one of the braids b_{l}, is reduced.

Proof. Suppose that the deletion of a letter i_{m} from i results in a word i′ that is not reduced. We will show that i_{m} must be a middle letter of one of the braids b_{l}. Denote by y the element of W represented by i_{m+1} ⋯ i_{n}, and note that y^{−1}(α_{i_{m}}) ∈ Φ(w). Since i is contracted, we need only show that y^{−1}(α_{i_{m}}) is the highest root of a contractible triple of w. By Proposition 2.2, it suffices to show that y^{−1}(α_{i_{m}}) is the highest root of a (not necessarily contractible) inversion triple of w.

There is an index p such that the expression i_{p+1} ⋯ i_{m−1}i_{m+1} ⋯ i_{n} is reduced and i_{p} ⋯ i_{m−1}i_{m+1} ⋯ i_{n} is not. Let x be the element of W represented by the word i_{p+1} ⋯ i_{m−1}, and let α be the root x^{−1}(α_{i_{p}}), which is positive since i_{p} ⋯ i_{m−1} is reduced.

Let γ_{1} = y^{−1}(α + α_{i_{m}}), γ_{2} = −y^{−1}(α) and γ_{3} = y^{−1}(α_{i_{m}}). Clearly, γ_{1} + γ_{2} = γ_{3}. We claim that this is an inversion triple of the element w′ represented by i_{p} ⋯ i_{m} ⋯ i_{n}. Since i_{p+1} ⋯ i_{m−1}i_{m+1} ⋯ i_{n} is reduced and i_{p} ⋯ i_{m−1}i_{m+1} ⋯ i_{n} is not, y^{−1}x^{−1}(α_{i_{p}}) is negative. This root equals y^{−1}(α), hence γ_{2} is positive.

Next, we show that γ_{1} is positive. Let c = B(α, α_{i_{m}}). Note that α ≠ α_{i_{m}} (to see this, recall from above that y^{−1}(α_{i_{m}}) is positive whereas y^{−1}(α) is negative). Therefore, since W is finite, c can only equal −1/2, 0, or 1/2. If c were to equal 0, then y^{−1}s_{i_{m}}(α) would equal y^{−1}(α), a negative root; this would contradict the fact that y^{−1}s_{i_{m}}(α) belongs to the inversion set of w. If c were to equal 1/2, then the positive root y^{−1}s_{i_{m}}(α) would equal y^{−1}(α − α_{i_{m}}) = y^{−1}(α) − y^{−1}(α_{i_{m}}), which is

SCHUBERT VARIETIES AND FREE BRAIDEDNESS 9

negative, another contradiction. Thus, c = −1/2, and this implies that the positive root y^{−1}s_{i_{m}}(α) equals y^{−1}(α + α_{i_{m}}) = γ_{1}.

We verify that the roots γ_{1} and γ_{2} are sent negative by w′. This follows from the calculations w′(γ_{1}) = s_{i_{p}}xs_{i_{m}}yy^{−1}(α + α_{i_{m}}) = s_{i_{p}}x(α) = −α_{i_{p}} and w′(γ_{2}) = −s_{i_{p}}xs_{i_{m}}yy^{−1}(α) = −s_{i_{p}}x(α + α_{i_{m}}) = −s_{i_{p}}(α_{i_{p}} + x(α_{i_{m}})) = α_{i_{p}} − s_{i_{p}}x(α_{i_{m}}). Regarding the latter, since α_{i_{p}} is simple and s_{i_{p}}x(α_{i_{m}}) is positive (because the word i_{p} ⋯ i_{m} is reduced), we have that w′(γ_{2}) is negative.

Thus, {γ_{1}, γ_{2}, γ_{3}} is an inversion triple of w′. Since every inversion triple of w′ is also an inversion triple of w, this proves that y^{−1}(α_{i_{m}}) is the highest root of some inversion triple of w, as desired. □

Remark 2.5. Let w ∈ W be freely braided. By [8, Proposition 3.1.3(ii)], any reduced expression for w can be transformed into a contracted expression by performing a sequence of short braid moves. Thus, in the case where W is finite, Proposition 2.4 can actually be used to decide whether the deletion of a letter from an arbitrary reduced expression for w results in another reduced expression.

3. Application to Schubert varieties

Throughout this section, we assume that our simply laced group W is the Weyl group of a semisimple simply-connected complex algebraic group G with fixed maximal torus T and Borel subgroup B ⊃ T. For each w ∈ W, let X_{w} be the associated Schubert variety, i.e., the closure of the cell BwB/B in the generalized flag variety G/B. Define, for each w ∈ W, the polynomial P_{w}(t) = ∑_{v⪯w} t^{ℓ(v)}, where ⪯ denotes the Bruhat–Chevalley partial order on W. Then P_{w}(t^{2}) is the Poincaré polynomial for the cohomology ring of X_{w}.

Carrell–Peterson have shown that a Schubert variety X_{w} is rationally smooth if and only if the polynomial P_{w}(t) is symmetric, meaning P_{w}(t) = t^{ℓ(w)}P_{w}(1/t) [4]. Note that in the simply laced setting, rational smoothness and smoothness are equivalent for Schubert varieties (this is an unpublished result of Peterson; see [5]).

10 R.M. GREEN AND J. LOSONCZY

Let w ∈ W. The set of letters from I appearing in some (any) reduced expression for w will be called the support of w, and will be denoted by supp(w). If w is freely braided, then it has a contracted reduced expression by Proposition 2.3. Consideration of this reduced expression reveals that ℓ(w) − N(w) ≥ # supp(w).

Definition 3.1. If w is freely braided and # supp(w) = ℓ(w) − N(w), then w is said to be content maximal.

Let v, w ∈ W and let i = i_{1} ⋯ i_{n} be a reduced expression for w. Recall that v ⪯ w if and only if v is represented by a (possibly empty) word of the form i_{μ_{1}} ⋯ i_{μ_{m}}, where 1 ≤ μ_{1} < ⋯ < μ_{m} ≤ n [10, Theorem 5.10].

Theorem 3.2. Let w ∈ W be freely braided. Then the Schubert variety X_{w} is smooth if and only if w is content maximal.

Proof. By Proposition 2.4 and [6, Lemma 2], there are exactly ℓ(w)−N(w) elements of W having length ℓ(w) − 1 that are less than w in the Bruhat–Chevalley order. If X_{w} is (rationally) smooth, then P_{w}(t) is symmetric, and it follows that there are exactly ℓ(w) − N(w) elements of length 1 that are less than w, as required.

Now assume that the support of w has ℓ(w) − N(w) elements.

Let i = u_{0}b_{1}u_{1}b_{2}u_{2} ⋯ b_{N(w)}u_{N(w)} be a contracted reduced expression for w. By content maximality, the only letters appearing more than once in i are the outer factors of the braids b_{l}. We will show that the polynomial P_{w}(t) is symmetric, and the theorem will follow from the results of Carrell and Peterson cited above.

We argue by induction on d = ℓ(w). For d ≤ 3, the polynomial P_{w}(t) is symmetric by direct verification, so we assume d > 3. There are two cases to consider.

Case 1: u_{N(w)} is a nonempty word, ending in the letter i.

Let i′ be the reduced expression obtained from i by deleting i, and let v be the group element represented by i′. Now, i appears exactly once in i, because it does not appear in a braid. Therefore, by [8, Lemma 3.2.2], v is freely braided and

SCHUBERT VARIETIES AND FREE BRAIDEDNESS 11

N(v) = N(w). We thus also have # supp(v) = ℓ(v) − N(v), so that v satisfies the inductive hypothesis.

Since i does not appear in i′, the set {x : x ⪯ w} equals the disjoint union of the sets L_{1} = {x : x ⪯ v} and L_{2} = {xs_{i} : x ⪯ L_{1}}. Furthermore, for x ∈ L_{1}, we have ℓ(xs_{i}) = ℓ(x) + 1. It follows that P_{w}(t) = (1 + t)P_{v}(t). Since deg P_{v}(t) = deg P_{w}(t) − 1 and P_{v}(t) is symmetric by induction, P_{w}(t) is symmetric.

Case 2: u_{N(w)} is empty.

Since d > 3, we have N(w) ≥ 1. Write b_{N(w)} = iji and let v = ws_{i}s_{j}s_{i}. Since i and j do not lie in the support of v, any element x ⪯ w is uniquely of the form x = ab where a ⪯ v, b ⪯ s_{i}s_{j}s_{i} and ℓ(ab) = ℓ(a) + ℓ(b).

It follows that P_{w}(t) = (1 + t)(1 + t + t^{2})P_{v}(t). By [8, Lemma 3.2.2] again (applied three times successively to w), v is freely braided and N(v) = N(w) − 1. Since ℓ(v) = ℓ(w) − 3 and # supp(v) = # supp(w) − 2, the inductive hypothesis is satisfied by v, so that P_{v}(t) is symmetric. Since deg P_{v}(t) = deg P_{w}(t) − 3, the induction is complete. □

Remark 3.3. Another possible approach to proving Theorem 3.2 would be to use the criterion for smoothness recently found by Billey–Postnikov [3].

An element w ∈ W is said to be fully commutative if every reduced expression for w can be transformed into any other by performing a sequence of short braid moves [14]. This is equivalent to the requirement that N(w) = 0 [8, Proposition 1.2.2]. Thus, every fully commutative element of W is freely braided. Note that the fully commutative elements coincide with the “short-braid avoiding” elements of [6] in the case where W is simply laced (the only case considered in this paper).

We have the following immediate corollary of Theorem 3.2, which was originally proved by Fan for short-braid avoiding elements of an arbitrary Weyl group [6, Proposition 3].

Corollary 3.4. Let w ∈ W be fully commutative. Then the Schubert variety X_{w} is smooth if and only if w equals a product of ℓ(w) distinct generators. □

4. Enumeration of content maximal elements

Motivated by Theorem 3.2, we derive generating functions for the number of content maximal elements in types A_{n}, D_{n} and E_{n}. In the E-series, we allow n to be any positive integer ≥ 6, since all groups in this series have finitely many freely braided elements [8, Theorem 3.3.3] and our methods apply equally well to them.

It turns out that all three generating functions are based on a single recurrence relation; we explain by using the following set-up. Consider a nested sequence of simply laced Coxeter groups W_{1} ⊂ W_{2} ⊂ ⋯ with generating sets S_{1} ⊂ S_{2} ⊂ ⋯ and alphabets I_{1} ⊂ I_{2} ⊂ · · · . Thus, for each n ≥ 1, we have S_{n} = {s_{i} : i ∈ I_{n}}. Assume that for any positive integers m < n and any i, j ∈ I_{m}, the order of s_{i}s_{j} is the same in W_{m} as it is in W_{n}. Assume further the existence of a sequence (i_{n})_{n≥1} such that each i_{n} lies in I_{n}, and whenever n > 1, we have m_{i_{n}j} = 2 for all j ∈ I_{n−1} except for j = i_{n−1}.

Some ad hoc terminology will be useful. Given a content maximal element w and a letter i, we say that i is not braided in w if i appears at most once in any reduced expression for w. If i appears more than once in some reduced expression for w, then by content maximality it appears exactly twice and there is precisely one letter j between the two occurrences of i such that m_{ij} = 3. In this case, we say that i is braided in w with j. It is clear that if i is braided (in w) with j, then j is braided with i. Note that if i is braided in w, then it is braided with precisely one other letter.

Let w ∈ W. In the proof of the following lemma, we will use a result of Matsumoto [13] and Tits [15], which states that any reduced expression for w can be transformed into any other by applying a sequence of long and short braid moves.

Denote by f(n) the number of content maximal elements in W_{n}.

Lemma 4.1. Let w ∈ W_{n} be content maximal. Then w satisfies one of the follow-

SCHUBERT VARIETIES AND FREE BRAIDEDNESS 13

ing seven mutually exclusive conditions:

(i) i_{n} ∉ supp(w); (ii) i_{n} ∈ supp(w) and i_{n} is not braided in w and if n > 1, then i_{n−1} ∉ supp(w); (iii) i_{n} ∈ supp(w) and i_{n} is not braided in w and i_{n−1} ∈ supp(w) and ℓ(ws_{i_{n}}) < ℓ(w); (iv) i_{n} ∈ supp(w) and i_{n} is not braided in w and i_{n−1} ∈ supp(w) and ℓ(s_{i_{n}}w) < ℓ(w);

(v) i_{n} ∈ supp(w) and i_{n} is braided in w and if n > 2, then i_{n−2} ∉ supp(w); (vi) i_{n} ∈ supp(w) and i_{n} is braided in w and i_{n−2} ∈ supp(w) and ℓ(ws_{i_{n−1}}) < ℓ(w); (vii) i_{n} ∈ supp(w) and i_{n} is braided in w and i_{n−2} ∈ supp(w) and ℓ(s_{i_{n−1}}w) < ℓ(w).

For n > 3, the number of content maximal elements of W_{n} satisfying the respective conditions (i), (ii), …, (vii) is f(n − 1), f(n − 2), f(n − 1) − f(n − 2), f(n − 1) − f(n − 2), f(n − 3), f(n − 2) − f(n − 3), and f(n − 2) − f(n − 3). We therefore have the recurrence f(n) = 3f(n − 1) + f(n − 2) − f(n − 3).

Note. In conditions (iii)–(v), we are assuming n > 1. In conditions (vi) and (vii), we are assuming n > 2.

Proof. Suppose that i_{n} ∈ supp(w) and i_{n} is not braided in w and i_{n−1} ∈ supp(w). We claim that either ℓ(ws_{i_{n}}) < ℓ(w) or ℓ(s_{i_{n}}w) < ℓ(w). Fix a contracted reduced expression for w. By our assumptions, i_{n} appears exactly once and i_{n−1} appears at least once and at most twice in this expression. If i_{n−1} appears only once, then, since m_{i_{n}j} = 2 for all j ∈ I_{n−1} except for j = i_{n−1}, we can commute i_{n} to one of the ends of our contracted expression. If instead i_{n−1} appears twice, then it is braided, necessarily with i_{n−2}. Once again, we can commute i_{n} to an end of our contracted expression. Thus, either (iii) or (iv) holds. Note that these conditions cannot hold simultaneously, by our assumption that i_{n} is not braided in w together with the result of Matsumoto and Tits cited above.

Similar reasoning shows that if i_{n} ∈ supp(w) and i_{n} is braided in w and i_{n−2} ∈ supp(w), then exactly one of the conditions (vi) and (vii) holds. Thus, our seven conditions are mutually exclusive and collectively exhaustive.

14 R.M. GREEN AND J. LOSONCZY

Regarding the cardinality assertions, let C_{n} denote the set of content maximal elements in W_{n} and let C_{n}(i), C_{n}(ii), …, C_{n}(vii) denote the respective subsets of elements satisfying conditions (i), (ii), …, (vii). We have C_{n}(i) = C_{n−1}, so that #C_{n}(i) = f(n − 1). The mapping C_{n}(ii) ⟶ C_{n−2} given by w ↦ ws_{i_{n}} is a well-defined bijection, hence #C_{n}(ii) = f(n − 2). Similarly, there are bijections C_{n}(iii) ⟶ C_{n−1} \ C_{n−2} and C_{n}(iv) ⟶ C_{n−1} \ C_{n−2}, given by w ↦ ws_{i_{n}} and w ↦ s_{i_{n}}w, respectively. It follows that #C_{n}(iii) = #C_{n}(iv) = f(n − 1) − f(n − 2). The remaining bijections are induced by left or right multiplication by s_{i_{n}}s_{i_{n−1}}s_{i_{n}}. □

Through direct computation, using either a variation of the proof of the above lemma or distinguished coset representatives for the subgroup W_{n−1} of W_{n}, one finds that the respective numbers of content maximal elements in Coxeter groups of type A_{1}, A_{2}, A_{3}, D_{4}, D_{5} and E_{6} are 2, 6, 19, 62, 201 and 652. This information, together with Lemma 4.1 itself, enables one to derive the ordinary generating functions for the numbers f(n) when W_{n} is of type A_{n} or D_{n} or E_{n}, as follows.

Theorem 4.2. The ordinary generating function for the number of content maximal elements is

\frac{2x − x^{3}}{1 − 3x − x^{2} + x^{3}} = 2x + 6x^{2} + 19x^{3} + 61x^{4} + 196x^{5} + ⋯  in type A_{n},
\frac{62x^{4} + 15x^{5} − 19x^{6}}{1 − 3x − x^{2} + x^{3}} = 62x^{4} + 201x^{5} + 646x^{6} + 2077x^{7} + ⋯ in type D_{n},
\frac{652x^{6} + 140x^{7} − 201x^{8}}{1 − 3x − x^{2} + x^{3}} = 652x^{6} + 2096x^{7} + 6739x^{8} + ⋯  in type E_{n}.

□

Acknowledgment

The authors thank the referees for their helpful and stimulating comments.

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