On Schubert varieties of complexity one
aa r X i v : . [ m a t h . A T ] O c t ON SCHUBERT VARIETIES OF COMPLEXITY ONE
EUNJEONG LEE, MIKIYA MASUDA, AND SEONJEONG PARK
Abstract.
Let B be a Borel subgroup of GL n ( C ) and T a maximal toruscontained in B . Then T acts on GL n ( C ) /B and every Schubert variety is T -invariant. We say that a Schubert variety is of complexity k if a maximal T -orbit in X w has codimension k . In this paper, we discuss topology, geometry,and combinatorics related to Schubert varieties of complexity one. Introduction
The flag manifold F ℓ ( C n ) is the homogeneous space GL n ( C ) /B , where B is theset of all upper triangular matrices in GL n ( C ). The left action of B on F ℓ ( C n ) hasfinitely many orbits BwB/B , where w is a permutation in S n , and the Schubertvariety X w is the (Zariski) closure of the B -orbit BwB/B . Most Schubert varietiesare not smooth and they are desingularized to Bott–Samelson varieties, see [2, 8].Let T be the set of all diagonal matrices in GL n ( C ). Then T is isomorphic tothe torus ( C ∗ ) n and acts on F ℓ ( C n ) by the left multiplication, and every Schubertvariety X w is a T -invariant irreducible subvariety of F ℓ ( C n ). We say that a Schubertvariety X w is of complexity k with respect to the action of T (or simply, X w is ofcomplexity k ) if a maximal T -orbit in X w has codimension k . In this paper, we areinterested in the Schubert varieties of complexity one.There were several studies on Schubert varieties of complexity zero (i.e., toricSchubert varieties) and related combinatorics. It is known that X w is of complexityzero if and only if a reduced decomposition of w consists of distinct letters, and inthis case X w is smooth and isomorphic to a Bott–Samelson variety ([10], [17]). Onthe other hand, permutation patterns are related to the form of reduced decompo-sitions. A reduced decomposition of w consists of distinct letters if and only if w avoids the patterns 321 and 3412, see [24]. It was also shown in [23] that a reduceddecomposition of w consists of distinct letters if and only if the Bruhat interval[ e, w ] is isomorphic to the Boolean algebra B ℓ ( w ) of rank ℓ ( w ), where ℓ ( w ) is thelength of w . The Bruhat interval polytope Q v,w , introduced in [25], is the convexhull of the points ( u (1) , . . . , u ( n )) in R n for all v ≤ u ≤ w . The Schubert variety X w is a smooth toric variety if and only if Q e,w is combinatorially equivalent to the ℓ ( w )-dimensional cube I ℓ ( w ) , see [21]. Theorem 1.1. [10, 17, 23, 24, 21]
The following are equivalent: (0) X w is a toric variety ( i.e., of complexity zero ) . (1) X w is a smooth toric variety. Date : October 9, 2020.2010
Mathematics Subject Classification.
Primary: 14M25, 14M15, secondary: 05A05.
Key words and phrases.
Schubert varieties, torus action, pattern avoidance, flag Bott–Samelson varieties, flag Bott maniflolds. (2) w avoids the patterns and . (3) A reduced decomposition of w consists of distinct letters. (4) X w is isomorphic to a Bott–Samelson variety. (5) The Bruhat interval [ e, w ] is isomorphic to B ℓ ( w ) , the Boolean algebra ofrank ℓ ( w ) . (6) The Bruhat interval polytope Q e,w is combinatorially equivalent to the ℓ ( w ) -dimensional cube. Recall that for a decomposition w = s i . . . s i ℓ of a permutation w , the Bott–Samelson variety Z w is defined by the orbit space of Bs i B × · · · × Bs i ℓ B by theright action of B ℓ in (2.2). Then Z w has an iterated C P -bundle structure because Bs i k B/B ∼ = C P . Note that Bs i k B/B = X s ik and Bs i k B/B ∼ = F ℓ ( C ) = C P . ABott–Samelson variety Z w is not a toric variety in general, but it is diffeomorphic toa toric variety, called a Bott manifold. We refer the reader to [13] for more details.In this article, we study an analog of the equivalent statements (1) ∼ (6) in The-orem 1.1 for Schubert varieties of complexity one. Whereas every toric Schubertvariety is smooth, not every Schubert variety of complexity one is smooth. Forexample, the Schubert varieties X and X are of complexity one, but X is smooth and X is singular.Jantzen [15] generalized the notion of Bott–Samelson varieties. For an orderedtuple of permutations ( w , . . . , w r ), the variety Z ( w ,...,w r ) is the orbit space of Bw B × · · · × Bw r B by the right action of B r in (4.1). Unfortunately, therewas no name for the variety Z ( w ,...,w r ) in [15], but now, it is called a generalizedBott–Samelson variety (cf. [4] and [22]). When X w i is a flag manifold for every i = 1 , . . . , r , the variety Z ( w ,...,w r ) is called a flag Bott–Samelson variety , see [11]. Theorem 1.2.
For a permutation w in S n , the following are equivalent: (1 ′ ) X w is smooth and of complexity one. (2 ′ ) w contains the pattern exactly once and avoids the pattern . (3 ′ ) There exists a reduced decomposition of w containing s i s i +1 s i as a factorand no other repetitions. (4 ′ ) X w is isomorphic to a flag Bott–Samelson variety Z ( w ,...,w r ) such that r = ℓ ( w ) − , w k = s j s j +1 s j for some ≤ k ≤ r , w i = s j i for i = k , and j , . . . , j k − , j k +1 , . . . , j r , j, j + 1 are pairwise distinct. (5 ′ ) The Bruhat interval [ e, w ] is isomorphic to S × B ℓ ( w ) − . (6 ′ ) The Bruhat interval polytope Q e,w is combinatorially equivalent to the prod-uct of the hexagon and the cube I ℓ ( w ) − . Theorem 1.3.
For a permutation w in S n , the following are equivalent: (1 ′′ ) X w is singular and of complexity one. (2 ′′ ) w contains the pattern exactly once and avoids the pattern . (3 ′′ ) There exists a reduced decomposition of w containing s i +1 s i s i +2 s i +1 as afactor and no other repetitions. (4 ′′ ) X w is isomorphic to a generalized Bott–Samelson variety Z ( w ,...,w r ) suchthat r = ℓ ( w ) − , w k = s j +1 s j s j +2 s j +1 for some ≤ k ≤ r , w i = s j i for i = k , and j , . . . , j k − , j k +1 , . . . , j r , j, j + 1 , j + 2 are pairwise distinct. (5 ′′ ) The Bruhat interval [ e, w ] is isomorphic to [ e, × B ℓ ( w ) − . (6 ′′ ) The Bruhat interval polytope Q e,w is combinatorially equivalent to the prod-uct of Q e, and the cube I ℓ ( w ) − . N SCHUBERT VARIETIES OF COMPLEXITY ONE 3
The equivalence between the first two statements (respectively, the second andthe third statements) in Theorems 1.2 and 1.3 is an immediate consequence of [19]and [24] (respectively, [7]). The following diagram shows how we prove the maintheorems in the paper. We prove Theorems 1.2 and 1.3 in parallel.(1 ′ ) (2 ′ ) (3 ′ )(4 ′ )(5 ′ )(6 ′ ) Proposition 3.7([19, 24]) Theorem 3.9([7]) T h e o r e m . Corollary 4.9 P r o p o s i t i o n . Theorem 5.8Like as a Bott–Samelson variety is diffeomorphic to a Bott manifold having ahigher rank torus action, a flag Bott–Samelson variety is diffeomorphic to a flagBott manifold which admits a higher rank torus action. Whereas a Bott manifold isa toric variety, a flag Bott manifold is not a toric variety in general, but it becomesa GKM manifold. In addition, we will see that every smooth Schubert variety ofcomplexity one is diffeomorphic to a flag Bott manifold.This paper is organized as follows. Section 2 contains basic notions and factsabout symmetric groups, Schubert varieties, Bott–Samelson varieties and Bott tow-ers. In Section 3, we introduce the various relation between the pattern avoidanceof a permutation and the complexity of a Schubert variety, and see the equivalenceamong the first three statements in Theorems 1.2 and 1.3. In Section 4, we in-troduce the notions of flag Bott–Samelson varieties and generalized Bott–Samelsonvarieties, and prove the implications (1 ′ ) ⇒ (4 ′ ) ⇒ (5 ′ ) and (1 ′′ ) ⇒ (4 ′′ ) ⇒ (5 ′′ ) inTheorems 1.2 and 1.3, respectively. In Section 5, we study the properties of Bruhatintervals and Bruhat interval polytopes related to Schubert varieties of complexityone, and then complete proofs of Theorems 1.2 and 1.3. In Section 6, we intro-duce the notion of flag Bott manifolds, and then show that every smooth Schubertvariety of complexity one is diffeomorphic to a flag Bott manifold.2. Preliminaries
In this section, we first prepare basic facts about symmetric groups and Schu-bert varieties from [3], and then see the relation among Schubert varieties, Bott–Samelson varieties, and Bott towers.Let G = GL n ( C ) and B the set of upper triangular matrices in G . We denoteby T the set of diagonal matrices in G . Then, T ∼ = ( C ∗ ) n . The homogeneous space G/B is a smooth projective variety can be identified with the set F ℓ ( C n ) = { ( { } ( V ( V ( · · · ( V n = C n ) | dim C V i = i for i = 1 , . . . , n } of chains of subspaces of C n . The Weyl group of G is identified with the symmetricgroup S n on the set [ n ] := { , , . . . , n } . For an element w ∈ S n , we use theone-line notation w = w (1) w (2) · · · w ( n ) . In this one-line notation, the identity element e is presented by e = 1 2 · · · n .We denote the set of transpositions by(2.1) T = { ( i, j ) | ≤ i < j ≤ n } , EUNJEONG LEE, MIKIYA MASUDA, AND SEONJEONG PARK which are permutations on [ n ] swapping i and j . The simple transpositions s i arethe transpositions of the form s i := ( i, i + 1) , for i = 1 , . . . , n − . Since S n is generated by simple transpositions, every w ∈ S n can be expressed as aproduct of simple transpositions. If w = s i · · · s i ℓ and ℓ is minimal among all suchexpressions, then ℓ is called the length of w (written ℓ ( w ) = ℓ ) and the expression s i · · · s i ℓ is called a reduced decomposition (or reduced expression or reduced word )for w . We denote by R ( w ) the set consisting of all reduced decompositions of w. Aconsecutive substring of a reduced decomposition is called a factor . For instance,since R (321) = { s s s , s s s } and R (3412) = { s s s s , s s s s } , no reduced decomposition for 3412 contains s i s i +1 s i as a factor unlike 321.The Bruhat order on S n is defined by v ≤ w if a reduced decomposition of v isa substring of some reduced decomposition of w . Then S n with the Bruhat order is a graded poset, with rank function given by length. Figure 1 gives the Hassediagram for the Bruhat order on S . Figure 1.
The Bruhat order on S .The complex torus T acts on G/B by the left multiplication, and the set of T -fixed points is identified with S n . More precisely, each element w ∈ S n correspondsto a coordinate flag given by( { } ( h e w (1) i ( h e w (1) , e w (2) i ( · · · ( V n = C n )where e , . . . , e n are the standard basis vectors in C n . We denote by wB thisstandard coordinate flag. It is well-known that F ℓ ( C n ) has a Bruhat decomposition F ℓ ( C n ) = G w ∈ S n BwB/B.
Moreover, the B -orbit BwB/B is isomorphic to C ℓ ( w ) and called a Schubert cell .The (Zariski) closure of
BwB/B is the
Schubert variety X w , and each Schubert N SCHUBERT VARIETIES OF COMPLEXITY ONE 5 variety decomposes into Schubert cells: X w = G v ≤ w BvB/B.
Note that most Schubert varieties are singular, and they are desingularized usingBott–Samelson varieties. Let w be a permutation and consider a decomposition w =( i , . . . , i ℓ ) of w (not necessarily reduced). The Bott–Samelson variety associatedwith w , denoted Z w , is the quotient of Bs i B × · · · × Bs i ℓ B by the action of B ℓ := B × · · · × B | {z } ℓ given by:(2.2) ( b , . . . , b ℓ ) · ( p , . . . , p ℓ ) := ( p b , b − p b , . . . , b − ℓ − p ℓ b ℓ )for ( b , . . . , b ℓ ) ∈ B ℓ and ( p , . . . , p ℓ ) ∈ Q ℓk =1 Bs i k B . Then Z w is smooth and ithas an iterated C P -bundle structure because Bs i k B/B ∼ = C P . Moreover, theleft multiplication of B on Bs i B induces an action of B on Z w and we have a B -equivariant map p w : Z w → G/B defined by p w ( p , . . . , p ℓ ) = p · · · p ℓ B . If w is a reduced decomposition of w , then p w gives a resolution of singularities for X w . See [2, 8] for details.A Bott–Samelson variety Z w is not a toric variety in general, but it is diffeomor-phic to a toric variety, called a Bott manifold. Definition 2.1. [13] A
Bott tower is an iterated C P -bundle: B ℓ B ℓ − · · · B B ,P ( C ⊕ ξ ℓ ) C P { a point } π ℓ π ℓ − π π where each B k is the complex projectivization of the Whitney sum of a holomorphicline bundle ξ k over B k − and the trivial bundle C . Each B k is called a Bott manifold (of height k ).Let γ j be the tautological line bundle over B j and γ j,i the pullback of γ j by theprojection π i ◦ · · · ◦ π j +1 : B i → B j for i > j . For convenience, we define γ j,j = γ j .Then for each k = 2 , . . . , ℓ , there exist a j,k ∈ Z for 1 ≤ j < k such that ξ k = O ≤ j Theorem 2.2 ([13, Proposition 3.10]) . For w ∈ S n , let w = ( i , i , . . . , i ℓ ) be areduced decomposition. Then the Bott–Samelson variety Z w is diffeomorphic to aBott manifold B ℓ determined by the integers (2.3) a j,k = h e i j − e i j +1 , e i k − e i k +1 i for ≤ j < k ≤ ℓ . Here, e , . . . , e n +1 are the standard basis vectors in R n +1 and h· , ·i is the standard inner product in R n +1 . Note that most Schubert varieties are neither smooth nor toric. However toricSchubert varieties are smooth and they are Bott manifolds. For a toric Schubertvariety X w , every reduced decomposition of w consists of distinct letters, and hencefor the associated Bott manifold the integers a j,k in (2.3) are either 0 or − Pattern avoidance and the complexity In this section, we define the complexity of a Schubert variety using the notionof complexity of a torus action, and see the relation between the complexity ofa Schubert variety and patterns of a permutation. We also show the equivalenceamong the first three statements in Theorems 1.2 and 1.3.Let X be a smooth complex projective algebraic variety having an action ofalgebraic torus T = ( C ∗ ) n . When the maximal T -orbit in X has codimension k , wecall the number k the complexity of the action.Every Schubert variety X w is a T -invariant irreducible subvariety of F ℓ ( C n ).Note that the T -fixed point set of X w is the set of coordinate flags uB for u ≤ w .That is, there is a bijection between ( X w ) T and the Bruhat interval[ e, w ] := { v ∈ S n | v ≤ w } . Using the Pl¨ucker embedding, we get a moment map µ : F ℓ ( C n ) → R n which sends uB ( u − (1) , . . . , u − ( n )) , and the image µ ( F ℓ ( C n )) is a simple convex polytopePerm n − := Conv { ( w (1) , . . . , w ( n )) ∈ R n | w ∈ S n } , called the permutohedron .The notion of Bruhat interval polytopes was introduced in [25] as a generalizationof the notion of permutohedra. For two elements v and w in S n with v ≤ w inBruhat order, the Bruhat interval polytope Q v,w is the convex polytope given bythe convex hull of the points ( u (1) , . . . , u ( n )) ∈ R n for v ≤ u ≤ w . Then for aSchubert variety X w , the moment map image µ ( X w ) becomes the Bruhat intervalpolytope Q e,w − := Conv { ( u (1) , . . . , u ( n )) ∈ R n | u ≤ w − } , and moreover, the images µ ( uB ) for all u ≤ w are the vertices of the Bruhat intervalpolytope Q e,w − . Proposition 3.1 (cf. [25, Proposition 6.20] and references therein) . The verticesof Q e,w − are the points ( u (1) , . . . , u ( n )) ∈ R n for all u ≤ w − . We refer the readers to [21] for more details on moment maps and the correspon-dence between Schubert varieties and Bruhat interval polytopes. For v, w in S n with v < w in Bruhat order, the Bruhat interval [ v, w ] is the subposet of( S n , < ) consisting of all the permutations u with v ≤ u ≤ w , and the Bruhat interval [ e, w ] is alsoknown as the principal order ideal of w , see [23]. N SCHUBERT VARIETIES OF COMPLEXITY ONE 7 Remark 3.2 (cf. [21, Remark 4.4]) . For every w ∈ S n with n ≤ Q v,w and Q v − ,w − are combinatorially equivalent. However, for w = 35412 in S , the Bruhatinterval polytopes Q e,w and Q e,w − are not combinatorially equivalent. In fact, their f -vectors are different: f ( Q e, ) = (60 , , , , 1) and f ( Q e, ) = (60 , , , , , so Q e,w has one more edge than Q e,w − . These vectors are computed by a computerprogram SageMath.Note that the complex dimension of a maximal T -orbit in X w is the same asthe real dimension of the moment map image µ ( X w ). Hence we can define the complexity of X w as follows: c ( w ) = dim C X w − dim R Q e,w − = ℓ ( w ) − dim R Q e,w − . See [21, Section 6] for more details. For example, c (3142) = 3 − c (4132) =4 − (1) Q e, − = Q e, . (2) Q e, − = Q e, . Figure 2. Moment map images of X and X .For a permutation w ∈ S n , the dimension of the polytope Q e,w is related to areduced decomposition of w . The support of w is the set of distinct letters appearingin a reduced decomposition of w , and we denote it by supp( w ). In fact, supp( w ) isthe same as the set of atoms in the Bruhat interval [ e, w ].It follows from [20, Corollary 7.13] that the dimension of a Bruhat intervalpolytope Q e,w is determined by the number of edges emanating from the vertex(1 , , . . . , n ), and moreover, by [20, Remark 7.5(5)], that number is the same as thecardinality of supp( w ). Indeed, the vertices connecting to the vertex (1 , , . . . , n )by an edge are presented by(1 , , . . . , i − , i + 1 , i, i + 2 , . . . , n ) i th EUNJEONG LEE, MIKIYA MASUDA, AND SEONJEONG PARK for all s i ∈ supp( w ). Therefore, the complexity of X w is determined by the supportof w : c ( w ) = ℓ ( w ) − | supp( w ) | . It means that c ( w ) equals the number of repeated letters in a reduced decompositionof w . In [24], Tenner denoted this quantity by rep( w ) and found a relation withpatterns in w . Definition 3.3. For w ∈ S n and p ∈ S k with k ≤ n , we say that the permutation w contains the pattern p if there exists a sequence 1 ≤ i < · · · < i k ≤ n such that w ( i ) · · · w ( i k ) is in the same relative order as p (1) · · · p ( k ). If w does not contain p ,then we say that w avoids p , or is p -avoiding .For example, the permutation 4231 in S has the pattern 321 twice. Let[321; 3412]( w ) be the number of distinct 321-and 3412-patterns in a permutation w .Then we can interpret Theorem 2.17 in [24] in terms of the complexity of X w . Theorem 3.4. [24, Theorem 2.17] For a permutation w in S n , we have(1) c ( w ) = 0 if and only if [321; 3412]( w ) = 0 , and(2) c ( w ) = 1 if and only if [321; 3412]( w ) = 1 . Example 3.5. For w ∈ S , [321; 3412]( w ) = 1 if and only if w is one of thefollowing permutations1432 , , , , , , . The first six permutations contains the 321-pattern once, and the last one avoidsit. Recall that not every Schubert variety is smooth, and Lakshmibai and Sandhyacharacterized the smoothness of Schubert varieties in terms of pattern avoidance. Theorem 3.6. [19] For a permutation w ∈ S n , the Schubert variety X w is smoothif and only if w avoids the patterns and . Combining Theorems 3.4 and 3.6, we obtain the following proposition. It showsthe equivalence between the first two statements in Theorems 1.2 and 1.3. Proposition 3.7. For w ∈ S n , the following hold:(1) X w is a toric variety if and only if w avoids the patterns both and ;(2) X w is smooth and of complexity one if and only if w has the pattern exactly once and avoids the pattern ; and(3) X w is singular and of complexity one if and only if w has the pattern exactly once and avoids the pattern . In general, c ( w ) ≤ [321; 3412]( w ) and the equality holds only when w avoidsevery pattern in the set { , , , , , , , , , } , see [24, Theorem 3.2]. For example, the permutation 4321 has four distinct 321-patterns but c (4321) = 3. In the set above, every pattern except 4321 has thepattern 3412. Since w avoids the pattern 3412 when X w is smooth, we get thefollowing proposition. Proposition 3.8. For a smooth Schubert variety X w , the complexity of X w is lessthan or equal to the number of distinct -patterns in w , and the equality holds ifand only if w avoids . N SCHUBERT VARIETIES OF COMPLEXITY ONE 9 Since c ( w ) equals the number of repeated letters in a reduced decompositionof w , Theorem 3.4(1) implies that a permutation w ∈ S n avoids both 321-and3412-patterns if and only if every reduced decomposition of w consists of distinctletters. Daly characterized the reduced decomposition of the permutations satisfy-ing [321; 3412]( w ) = 1 as follows. Theorem 3.9. [7] For a permutation w ∈ S n , the following hold:(1) w contains exactly one pattern and avoids if and only if thereexists a reduced decomposition of w containing s i s i +1 s i as a factor and noother repetitions.(2) w contains exactly one pattern and avoids if and only if thereexists a reduced decomposition of w containing s i +1 s i s i +2 s i +1 as a factorand no other repetitions. The above theorem shows the equivalence between the second and the thirdstatements in Theorems 1.2 and 1.3.It is also shown in [6] that there is a one-to-one correspondence between { w ∈ S n | w contains exactly one 321 pattern and avoids 3412 } and { w ∈ S n +1 | w contains exactly one 3412 pattern and avoids 321 } , and the cardinality of the set is given in A001871 in OEIS [1]. Thus we obtain that { X w ⊆ F ℓ ( C n ) | X w is smooth and of complexity one } = { X w ′ ⊆ F ℓ ( C n +1 ) | X w ′ is singular and of complexity one } . Flag Bott–Samelson varieties andgeneralized Bott–Samelson varieties In this section, we recall flag Bott–Samelson varieties from [11] and generalizedBott–Samelson varieties from [15].Let G = GL n ( C ). For a subset I ⊆ [ n ] := { , . . . , n } , we define the subgroup W I of W by W I := h s i | i ∈ I i . Then the parabolic subgroup P I of G corresponding to I is defined by P I = Bw I B ⊆ G, where w I is the longest element in W I . Definition 4.1 ([11, Definition 2.1]) . Let I = ( I , . . . , I r ) be a sequence of subsetsof [ n ]. The flag Bott–Samelson variety Z I is defined by the orbit space Z I := ( P I × · · · × P I r ) / Θ , where the right action Θ of B r := B × · · · × B | {z } r on Q rk =1 P I k is defined by(4.1) Θ(( p , . . . , p r ) , ( b , . . . , b r )) = ( p b , b − p b , . . . , b − r − p r b r )for ( p , . . . , p r ) ∈ Q rk =1 P I k and ( b , . . . , b r ) ∈ B r . Note that P I k /B ∼ = F ℓ ( C | I k | ) for k = 1 , . . . , r . Hence if | I | = · · · = | I r | = 1,then Z I becomes a Bott–Samelson variety. Each Bott–Samelson variety can bedescribed as an iterated C P -bundle, whereas each flag Bott–Samelson variety canbe described as an iterated bundle whose fiber at each stage is a flag manifold.We recall properties of flag Bott–Samelson varieties from [11, Proposition 2.3].The flag Bott–Samelson variety Z I is a smooth projective variety and admits a nicedecomposition of affine cells. For ( w , . . . , w r ) ∈ Q rk =1 W I k , we define Z ′ ( w ,...,w r ) and Z ( w ,...,w r ) in Z I by Z ′ ( w ,...,w r ) := ( Bw B × · · · × Bw r B ) / Θ ,Z ( w ,...,w r ) := ( Bw B × · · · × Bw r B ) / Θ . We call Z ( w ,...,w r ) a generalized Bott–Samelson variety , see [15] and also [4, 22].Then Z ′ ( w ,...,w r ) is an open dense subset of the generalized Bott–Samelson variety Z ( w ,...,w r ) and we have that Z ′ ( w ,...,w r ) ≃ C P rk =1 ℓ ( w k ) . For w k ∈ W I k , since Bw k B = F v ∈ WIv ≤ wk BvB , we have that(4.2) Z ( w ,...,w r ) = G ( v ,...,vr ) ∈ Q rk =1 WIkvk ≤ wk for k =1 ,...,r Z ′ ( v ,...,v r ) . Note that Z I = Z ( w I ,...,w Ir ) , where I = ( I , . . . , I r ) and w I k is the longest elementin W I k for k = 1 , . . . , r . Example 4.2. Let I = ( { , } , { } ). Then we have that Z I = Z ′ ( e,e ) ⊔ Z ′ ( s ,e ) ⊔ Z ′ ( s ,e ) ⊔ Z ′ ( s s ,e ) ⊔ Z ′ ( s s ,e ) ⊔ Z ′ ( s s s ,e ) ⊔ Z ′ ( e,s ) ⊔ Z ′ ( s ,s ) ⊔ Z ′ ( s ,s ) ⊔ Z ′ ( s s ,s ) ⊔ Z ′ ( s s ,s ) ⊔ Z ′ ( s s s ,s ) . Each Z ′ ( w ,w ) is isomorphic to an affine cell as follows: Z ′ ( e,e ) ≃ C , Z ′ ( s ,e ) ≃ Z ′ ( s ,e ) ≃ Z ′ ( e,s ) ≃ C ,Z ′ ( s s ,e ) ≃ Z ′ ( s s ,e ) ≃ Z ′ ( s ,s ) ≃ Z ′ ( s ,s ) ≃ C ,Z ′ ( s s s ,e ) ≃ Z ′ ( s s ,s ) ≃ Z ′ ( s s ,s ) ≃ C , Z ′ ( s s s ,s ) ≃ C . The multiplication map p I : Z I → G/B, [ p , . . . , p r ] p · · · p r B is well-defined because of the definition of the right action Θ. The maximal torus T acts on a generalized Bott–Samelson variety Z ( w ,...,w r ) by t · [ g , . . . , g r ] = [ tg , g , . . . , g r ] . We note that the multiplication map p I is T -equivariant. Proposition 4.3 ([11, Proposition 2.7]) . Let ( w , . . . , w r ) ∈ Q rk =1 W I k . Supposethat w · · · w r = v and ℓ ( w ) + · · · + ℓ ( w r ) = ℓ ( v ) . Then the multiplication map p I induces a birational morphism: p I | Z ( w ,...,wr ) : Z ( w ,...,w r ) → X v . Indeed, we have an isomorphism between dense open subsets: Z ′ ( w ,...,w r ) ∼ −→ BvB/B. N SCHUBERT VARIETIES OF COMPLEXITY ONE 11 Example 4.4. Let G = GL ( C ), and let I = ( { } , { } , { } ). Then the flag Bott–Samelson variety Z I has the decomposition: Z I = Z ′ ( e,e,e ) ⊔ Z ′ ( s ,e,e ) ⊔ Z ′ ( e,s ,e ) ⊔ Z ′ ( s ,s ,e ) ⊔ Z ′ ( e,e,s ) ⊔ Z ′ ( s ,e,s ) ⊔ Z ′ ( e,s ,s ) ⊔ Z ′ ( s ,s ,s ) . By Proposition 4.3, the multiplication map p I induces the isomorphism Z ′ ( w ,w ,w ) ∼ = Bw w w B/B except for ( w , w , w ) = ( s , e, s ). Moreover, one can see that themultiplication map p I is injective on Z I \ ( Z ′ ( s ,e,s ) ⊔ Z ′ ( s ,e,e ) ⊔ Z ′ ( e,e,s ) ). Corollary 4.5. Suppose that I , . . . , I r are pairwise disjoint subsets of [ n ] . Thenthe multiplication map p I with I = ( I , . . . , I r ) induces an isomorphism between Z ( w ,...,w r ) and its image X w ··· w r as T -varieties, where w k is an element in W I k .Proof. We first note that X w ··· w r = F v ≤ w ··· w r BvB/B . Since I , . . . , I r are pair-wise disjoint and w k ∈ W I k , we have that v ≤ w · · · w r ⇐⇒ v = v · · · v r , and v k ≤ w k in W I k . Therefore, we get that(4.3) X w ··· w r = G ( v ,...,vr ) ∈ Q rk =1 WIkvk ≤ wk for k =1 ,...,r Bv · · · v r B/B. This implies that the Schubert variety X w ··· w r can be decomposed into affine cellsindexed by elements in Q rk =1 [ e, w k ], where [ e, w k ] is a Bruhat interval in W I k .Moreover, since I , . . . , I r are pairwise disjoint, we know that ℓ ( v · · · v r ) = ℓ ( v ) + · · · + ℓ ( v r ). Hence by Proposition 4.3, the multiplication map p I induces a T -equivariant isomorphism Z ′ ( v ,...,v r ) ∼ = Bv · · · v r B/B for each ( v , . . . , v r ) ∈ Q rk =1 W I k . Hence by combining the above isomorphism withequations (4.2) and (4.3), the result follows. (cid:3) Example 4.6. Continuing Example 4.2, the multiplication map p I induces theisomorphisms: Z ′ ( e,e ) ∼ = BeB/B, Z ′ ( s ,e ) ∼ = Bs B/B,Z ′ ( s ,e ) ∼ = Bs B/B, Z ′ ( s s ,e ) ∼ = Bs s B/B,Z ′ ( s s ,e ) ∼ = Bs s B/B, Z ′ ( s s s ,e ) ∼ = Bs s s B/B,Z ′ ( e,s ) ∼ = Bs B/B, Z ′ ( s ,s ) ∼ = Bs s B/B,Z ′ ( s ,s ) ∼ = Bs s B/B, Z ′ ( s s ,s ) ∼ = Bs s s B/B,Z ′ ( s s ,s ) ∼ = Bs s s B/B, Z ′ ( s s s ,s ) ∼ = Bs s s s B/B Since we have the decomposition X w = F v ≤ w BvB/B , we get an isomorphism Z I = Z ( s s s ,s ) ∼ = X s s s s . Indeed, { , } ∩ { } = ∅ . Theorem 4.7. Every Schubert variety X w of complexity one is T -equivariantlyisomorphic to a generalized Bott–Samelson variety. If X w is of complexity one andsmooth, then it is T -equivariantly isomorphic to a flag Bott–Samelson variety.Proof. By Proposition 3.7 and Theorem 3.9, if X w is smooth, then there exists areduced decomposition w = s i · · · s i ℓ of w containing s i s i +1 s i as a factor and noother repetitions. On the other hand, if X w is singular, then there exists a reduced decomposition w = s i · · · s i ℓ of w containing s i +1 s i s i +2 s i +1 as a factor and noother repetitions. Hence there is a reduced decomposition w such that(1) if X w is smooth, then ( i q , i q +1 , i q +2 ) = ( i, i + 1 , i ) for some 1 ≤ q ≤ ℓ − X w is singular, then ( i q , i q +1 , i q +2 , i q +3 ) = ( i + 1 , i, i + 2 , i + 1) for some1 ≤ q ≤ ℓ − I = ( I , . . . , I r ) as follows:(1) If X w is smooth, then r = ℓ − I k = { i k } if 1 ≤ k < q, { i, i + 1 } if k = q, { i k +2 } if k > q. (2) If X w is singular, then r = ℓ − I k = { i k } if 1 ≤ k < q, { i, i + 1 , i + 2 } if k = q, { i k +3 } if k > q. Then, in any case, the subsets I , . . . , I r are pairwise disjoint. Moreover, if X w issmooth, then the concatenation of longest elements in W I k is the same as w . Henceby Corollary 4.5, the Schubert variety X w is T -equivariantly isomorphic to the flagBott–Samelson variety Z I .If X w is singular, then we set w k = s i k if 1 ≤ k < q,s i +1 s i s i +2 s i +1 if k = q,s i k +3 if q < k ≤ r. Then ( w , . . . , w r ) ∈ Q rk =1 W I k and w = w . . . w r . Hence by Corollary 4.5, X w is T -equivariantly isomorphic to the generalized Bott–Samelson variety Z ( w ,...,w r ) . (cid:3) The above theorem proves the implications (1 ′ ) ⇒ (4 ′ ) and (1 ′′ ) ⇒ (4 ′′ ) inTheorems 1.2 and 1.3, respectively.The following proposition can be checked easily and we omit the proof. Proposition 4.8. There is a natural bijection between the set of T -fixed points in Z ( w ,...,w r ) and the product of Bruhat intervals [ e, w ] × · · · × [ e, w r ] . If a Schubert variety X w and a generalized Bott–Samelson variety Z ( w ,...,w r ) areisomorphic as T -varieties, then they have the same set of T -fixed points. Hence weget the following. Corollary 4.9. If a Schubert variety X w is isomorphic to a generalized Bott–Samelson variety Z ( w ,...,w r ) as T -varieties, then the Bruhat interval [ e, w ] is iso-morphic to the product of Bruhat intervals Q rk =1 [ e, w k ] . The above corollary shows the implications (4 ′ ) ⇒ (5 ′ ) and (4 ′′ ) ⇒ (5 ′′ ) inTheorems 1.2 and 1.3, respectively. N SCHUBERT VARIETIES OF COMPLEXITY ONE 13 Bruhat intervals and Bruhat interval polytopes In this section, we study the properties of the Bruhat interval [ e, w ] and theBruhat interval polytope Q e,w for a permutation w with c ( w ) ≤ 1. We will completeproofs of Theorems 1.2 and 1.3.Recall that the following seven permutations in S , , , , , , w ) = 1 (in Example 3.5). One can easily check that • If w is either 1432 or 3214, then the Bruhat interval [ e, w ] is isomorphic to S and the Bruhat interval polytope Q e,w is combinatorially equivalent tothe hexagon Perm . • If w is one of the permutations 4132 , , e, w ] is isomorphic to S × B and the Bruhat interval polytope Q e,w is combinatorially equivalent to the hexagonal prism, Perm × I . • If w = 3412, then the Bruhat interval [ e, w ] and Q e,w are given in Fig-ure 3. Thus [ e, w ] is isomorphic to neither S × B nor B , and Q e,w iscombinatorially equivalent to neither the hexagonal prism nor a 4-cube.Here, B ℓ denotes the Boolean algebra of length ℓ . The above situation happens ingeneral and we have the following lemma that is obvious but plays a role in ourargument. Lemma 5.1. For any positive integer i , the following hold:(1) if w = s i s i +1 s i , then [ e, w ] is isomorphic to S as a poset, and Q e,w iscombinatorially equivalent to the hexagon Perm ; and(2) if w = s i +1 s i s i +2 s i +1 , then [ e, w ] is isomorphic to [ e, as a poset, and Q e,w is combinatorially equivalent to Q e, . Figure 3. The Bruhat interval [ e, Q e, . Therefore, if w = s i +1 s i s i +2 s i +1 for some positive integer i , then [ e, w ] is isomor-phic to neither S × B nor B , and Q e,w is combinatorially equivalent to neitherthe hexagonal prism nor a 4-cube. Proposition 5.2. For a permutation w in S n , the following hold:(1) If the Bruhat interval [ e, w ] is isomorphic to S × B ℓ ( w ) − ( as a poset ) ,then X w is smooth and of complexity one.(2) If the Bruhat interval [ e, w ] is isomorphic to [ e, × B ℓ ( w ) − ( as a poset ) ,then X w is singular and of complexity one. Note that the converse of each statement in the above proposition is also trueby Theroem 4.7 and Corollary 4.9. Proof of Proposition 5.2. Let us prove the first statement. Assume that the Bruhatinterval [ e, w ] is isomorphic to a poset S × B ℓ − . Note that(1) S × B ℓ − has ℓ − S × B ℓ − has rank ℓ ; and(3) every subinterval of S × B ℓ − is isomorphic to S × B k ( k < ℓ − 3) or B k ′ ( k ′ ≤ ℓ − c ( w ) = 1. If X w is not smooth, then w con-tains 3412 exactly once and avoids 321. Then by Theorem 3.9, there is a reduceddecomposition w of w such that w = s i · · · s i k − s i k +1 s i k s i k +2 s i k +1 s i k +3 · · · s i ℓ − and i k +1 = i k + 1 , i k +2 = i k + 2 . However, the interval [ e, s i k +1 s i k s i k +2 s i k +1 ] is a subinterval of neither S × B ℓ − nor B ℓ by (3). Therefore, X w is smooth.Now we prove the second statement. Note that [ e, × B ℓ − and S × B ℓ − .Since we already showed that X w is smooth and c ( w ) = 1 if and only if [ e, w ] isisomorphic to S × B ℓ − in the above, it is enough to show that c ( w ) = 1. Sincethe poset [ e, × B ℓ − has ℓ − ℓ , if [ e, w ] is isomorphicto [ e, × B ℓ − , then we get c ( w ) = 1. This proves the proposition. (cid:3) Therefore, the first five statements in Theorems 1.2 and 1.3 are equivalent.Now we determine the combinatorial type of the Bruhat interval polytope Q e,w when c ( w ) = 1. The combinatorial aspects of Bruhat interval polytopes are well-studied in [25]. Every face of a Bruhat interval polytope is itself a Bruhat intervalpolytope. However, for a subinterval [ x, y ] of an interval [ v, w ], Q x,y may not be aface of Q v,w . For a subinterval [ x, y ] of [ v, w ], we introduce a directed graph G v,wx,y which will be used to determine whether Q x,y is a face of Q v,w .Let v ≤ w in S n . For u ∈ [ v, w ], we define T ( u, [ v, w ]) := { ( i, j ) ∈ T | u < u ( i, j ) ≤ w, ℓ ( u ( i, j )) − ℓ ( u ) = 1 } ,T ( u, [ v, w ]) := { ( i, j ) ∈ T | v ≤ u ( i, j ) < u, ℓ ( u ) − ℓ ( u ( i, j )) = 1 } , where T is the set of transpositions in S n , see (2.1). We first construct a labelledgraph G x,y on [ n ] = { , . . . , n } having an edge between the vertices a and b if andonly if ( a, b ) ∈ T ( x, [ x, y ]). Theorem 5.3. [25] Let [ x, y ] ⊆ [ v, w ] . We define the graph G v,wx,y as follows:(1) The vertices of G v,wx,y are { , , . . . , n } , with vertices i and j identified if theyare in the same connected component of the graph G x,y . N SCHUBERT VARIETIES OF COMPLEXITY ONE 15 (2) There is a directed edge i → j if ( i, j ) ∈ T ( y, [ v, w ]) .(3) There is a directed edge j → i if ( i, j ) ∈ T ( x, [ v, w ]) .Then the Bruhat interval polytope Q x,y is a face of the Bruhat interval polytope Q v,w if and only if the graph G v,wx,y is a directed acyclic graph. Remark 5.4. The above theorem used Lemma 4.18 in [25]. Unfortunately, theproof of the lemma had a gap. Recently, the gap has been corrected by Caselli,D’Adderio and Marietti, see [5, Remarks 5.5]. Example 5.5. Let w = 1432, x = 1324, and y = 1342. Then we have T ( x, [ x, y ]) = { (3 , } , T ( y, [ e, w ]) = { (2 , } , and T ( x, [ e, w ]) = { (2 , } . Hence G e,wx,y is notacyclic, and thus Q x,y is not a face of Q e,w . See Figure 4.1 2 3 4 (1) G , . , (2) G , , . Figure 4. Examples of G x,y and G v,wx,y .We will use the following fact. Let a, b ∈ S n which do not have s r in theirsupports. Then we have that(5.1) a ≤ b ⇐⇒ s r a ≤ s r b ⇐⇒ as r ≤ bs r . In particular,(5.2) ℓ ( b ) − ℓ ( a ) = ℓ ( s r b ) − ℓ ( s r a ) = ℓ ( bs r ) − ℓ ( as r ) . Hence(5.3) T ( a, [ a, b ]) = T ( s r a, [ s r a, s r b ]) . For b w = s r w or ws r , in order to see which subinterval of [ v, b w ] gives a face of Q v, b w ,we prepare the following lemma. Lemma 5.6. Let u, v, w ∈ S n with v ≤ u ≤ w and s r / ∈ supp( w ) . Then(1) T ( u, [ v, s r w ]) = T ( u, [ v, w ]) and T ( u, [ v, ws r ]) = T ( u, [ v, w ]) ;(2) T ( u, [ v, s r w ]) = T ( u, [ v, w ]) ∪ { ( u − ( r ) , u − ( r + 1)) } ;(3) T ( u, [ v, ws r ]) = T ( u, [ v, w ]) ∪ { ( r, r + 1) } ;(4) T ( s r u, [ v, s r w ]) = T ( u, [ v, w ]) ;(5) T ( s r u, [ v, s r w ]) = T ( u, [ v, w ]) ∪ { ( u − ( r ) , u − ( r + 1)) } ;(6) T ( us r , [ v, ws r ]) = { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T ( u, [ v, w ]) } ; and(7) T ( us r , [ v, ws r ]) = { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T ( u, [ v, w ]) } ∪ { ( r, r + 1) } .Proof. The first statement is clear from the definition of T ( u, [ v, w ]) and the as-sumption s r / ∈ supp( w ). In (2) ∼ (5), the inclusion ( ⊇ ) is clear and hence it isenough to check the inclusion ( ⊆ ).In (2) and (3), for ( i, j ) ∈ T ( u, [ v, s r w ]) (respectively, ( i, j ) ∈ T ( u, [ v, ws r ])),if s r ∈ supp( u ( i, j )), then u ( i, j ) = s r u = u ( u − ( r ) , u − ( r + 1)) (respectively, u ( i, j ) = us r = u ( r, r + 1)); otherwise, ( i, j ) ∈ T ( u, [ v, w ]). This proves (2) and (3). In (4) and (5), we use (5.1) and (5.2). For ( i, j ) ∈ T ( s r u, [ v, s r w ]) in (4), since s r u < s r u ( i, j ) ≤ s r w and ℓ ( s r u ( i, j )) − ℓ ( s r u ) = 1, we have u < u ( i, j ) ≤ w and ℓ ( u ( i, j )) − ℓ ( u ) = 1. This proves that ( i, j ) ∈ T ( u, [ v, w ]). In (5), if( i, j ) ∈ T ( s r u, [ v, s r w ]), then s r u ( i, j ) = u or s r v ≤ s r u ( i, j ) < s r u . Note that s r u ( u − ( r ) , u − ( r +1)) = u . Thus if s r u ( i, j ) = u , then ( i, j ) = ( u − ( r ) , u − ( r +1));otherwise, ( i, j ) ∈ T ( u, [ v, w ]).Let us prove (6). It follows from (5.1) and (5.2) that there exists z ∈ [ us r , ws r ]such that ℓ ( z ) − ℓ ( us r ) = 1 if and only if there exists z ′ ∈ [ u, w ] such that z = z ′ s r and ℓ ( z ′ ) − ℓ ( u ) = 1. Here, z ′ is of the form z ′ = u ( i, j ) for some ( i, j ) ∈ T ( u, [ v, w ]).Since z = u ( i, j ) s r = us r ( s r ( i ) , s r ( j )), we get T ( us r , [ v, ws r ]) ⊇ { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T ( u, [ v, w ]) } . On the other hand, if z ′ s r = us r ( i ′ , j ′ ) for ( i ′ , j ′ ) ∈ T ( us r , [ v, ws r ]), then z ′ = us r ( i ′ , j ′ ) s r = u ( s r ( i ′ ) , s r ( j ′ )) and z ′ ≤ w . Hence there exists ( i, j ) ∈ T ( u, [ v, w ])such that ( i ′ , j ′ ) = ( s r ( i ) , s r ( j )). This proves (6).Finally, we prove (7). If ( i, j ) ∈ T ( us r , [ v, ws r ]), then us r ( i, j ) = u or vs r ≤ us r ( i, j ) < us r . If us r ( i, j ) = u , then ( i, j ) = ( r, r + 1). If vs r ≤ us r ( i, j ) < us r ,then there exists u ′ ∈ [ v, w ] such that us r ( i, j ) = u ′ s r and u ′ < u . Then u ′ = us r ( i, j ) s r = u ( s r ( i ) , s r ( j )). Note that ℓ ( u ) − ℓ ( u ′ ) = ℓ ( us r ) − ℓ ( u ′ s r ) = ℓ ( us r ) − ℓ ( us r ( i, j )) = 1 . Hence ( s r ( i ) , s r ( j )) ∈ T ( u, [ v, w ]). Thus, we get T ( us r , [ v, ws r ]) ⊆ { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T ( u, [ v, w ]) } ∪ { ( r, r + 1) } . On the other hand, ( r, r + 1) ∈ T ( us r , [ v, ws r ]) clearly. For ( i, j ) ∈ T ( u, [ v, w ]),since v ≤ u ( i, j ) < u and ℓ ( u ) − ℓ ( u ( i, j )) = 1, we get vs r ≤ u ( i, j ) s r < us r and ℓ ( us r ) − ℓ ( u ( i, j ) s r ) = 1 by (5.1) and (5.2). Note that u ( i, j ) s r = us r ( s r ( i ) , s r ( j )).Therefore, ( s r ( i ) , s r ( j )) ∈ T ( us r , [ v, ws r ]) . This proves (7). (cid:3) Proposition 5.7. For v, w in S n with v ≤ w , if s r supp( w ) , then both Q v,s r w and Q v,ws r are combinatorially equivalent to the polytope Q v,w × I .Proof. Note that for u ∈ S n with u ≤ s r w (respectively, u ≤ ws r ), if s r ∈ supp( u ),then there is a unique u ′ such that s r u ′ = u (respectively, u ′ s r = u ) because s r / ∈ supp( w ). We set˜ u = ( u if s r / ∈ supp( u ) ,u ′ if s r ∈ supp( u ) and b w = s r w or ws r . Then what we have to prove is that for any [ x, y ] ⊆ [ v, b w ], Q x,y is a face of Q v, b w if and only if Q ˜ x, ˜ y is a face of Q v,w ,which is equivalent to( ∗ ) G v, b wx,y is acyclic if and only if G v,w ˜ x, ˜ y is acyclicby Theorem 5.3. We will prove ( ∗ ) in the following.Note that for each [ x, y ] ⊆ [ v, b w ], there are three possibilities:(i) (˜ x, ˜ y ) = ( x, y ), i.e., s r supp( y ),(ii) (˜ x, ˜ y ) = ( x ′ , y ′ ), i.e., s r ∈ supp( x ), and(iii) (˜ x, ˜ y ) = ( x, y ′ ), i.e., s r supp( x ) but s r ∈ supp( y ). N SCHUBERT VARIETIES OF COMPLEXITY ONE 17 Since s r supp( w ), the graph G ˜ x, ˜ y has no directed edge between [ r ] and [ n ] \ [ r ].We prove ( ∗ ) by showing that G v, b wx,y is the directed graph G v,w ˜ x, ˜ y with one directededge added between [ r ] and [ n ] \ [ r ] in (i) and (ii) and with a pair of vertices in [ r ]and [ n ] \ [ r ] identified in (iii). Case 1: b w = s r w . Note that in cases (i) and (ii), T ( x, [ x, y ]) = T (˜ x, [˜ x, ˜ y ])by (5.3). Hence the vertex set of G v, b wx,y is the same as that of G v,w ˜ x, ˜ y . In case (i), itfollows from (1) and (2) of Lemma 5.6 that T ( x, [ v, s r w ]) = T (˜ x, [ v, w ])and T ( y, [ v, s r w ]) = T (˜ y, [ v, w ]) ∪ { ( y − ( r ) , y − ( r + 1)) } . Hence the graph G v, b wx,y is obtained from G v,w ˜ x, ˜ y by adding a directed edge from y − ( r ) ∈ [ r ] to y − ( r + 1) ∈ [ n ] \ [ r ]. In case (ii), it follows from (4) and (5)of Lemma 5.6 that T ( y, [ v, s r w ]) = T (˜ y, [ v, w ])and T ( x, [ v, s r w ]) = T (˜ x, [ v, w ]) ∪ { ((˜ x ) − ( r ) , (˜ x ) − ( r + 1)) } . Hence the graph G v, b wx,y is obtained from G v,w ˜ x, ˜ y by adding a directed edge from(˜ x ) − ( r + 1) ∈ [ n ] \ [ r ] to (˜ x ) − ( r ) ∈ [ r ]. In case (iii), by (1) and (4) of Lemma 5.6,we have T ( x, [ v, s r w ]) = T (˜ x, [ v, w ]) and T ( y, [ v, s r w ]) = T (˜ y, [ v, w ]) . Since T ( x, [ x, y ]) = T (˜ x, [˜ x, ˜ y ]) ∪ { ( x − ( r ) , x − ( r + 1)) } by Lemma 5.6(2), the graph G v, b wx,y is obtained from G v,w ˜ x, ˜ y by identifying the vertices x − ( r ) ∈ [ r ] and x − ( r + 1) ∈ [ n ] \ [ r ]. This proves Case 1. Case 2: b w = ws r . In case (i), since T ( x, [ x, y ]) = T (˜ x, [˜ x, ˜ y ]), the vertex set of G v, b wx,y is the same as that of G v,w ˜ x, ˜ y . It follows from (1) and (3) of Lemma 5.6 that T ( x, [ v, ws r ]) = T (˜ x, [ v, w ])and T ( y, [ v, ws r ]) = T (˜ y, [ v, w ]) ∪ { ( r, r + 1) } . Hence the graph G v, b wx,y is obtained from G v,w ˜ x, ˜ y by adding a directed edge from r to r + 1. In case (ii), since ˜ x ( i, j ) s r = ˜ xs r ( s r ( i ) , s r ( j )) = x ( s r ( i ) , s r ( j )), we get T ( x, [ x, y ]) = { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T (˜ x, [˜ x, ˜ y ]) } similarly to the proof of (6) in Lemma 5.6. Hence G x,y is obtained from G ˜ x, ˜ y byinterchanging the labelling of the vertices r and r + 1. By (6) and (7) of Lemma 5.6,we have T ( y, [ v, ws r ]) = { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T (˜ y, [ v, w ]) } and T ( x, [ v, ws r ]) = { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T (˜ x, [ v, w ]) } ∪ { ( r, r + 1) } . Thus the graph G v, b wx,y is obtained from G v,w ˜ x, ˜ y by interchanging the labelling of thevertices r and r + 1 and then adding a directed edge from r + 1 to r . In case (iii),it follows from (1) and (6) of Lemma 5.6 that T ( x, [ v, ws r ]) = T (˜ x, [ v, w ]) and T ( y, [ v, ws r ]) = { ( s r ( i ) , s r ( j )) | ( i, j ) ∈ T (˜ y, [ v, w ]) } . Note that T ( x, [ x, y ]) = T (˜ x, [˜ x, ˜ y ]) ∪ { ( r, r + 1) } by Lemma 5.6(3). Hence the graph G x,y is obtained from G ˜ x, ˜ y by adding an edgebetween the vertices r and r + 1. Therefore, the graph G v, b wx,y is obtained from G v,w ˜ x, ˜ y by identifying the vertices r and r + 1. (cid:3) The above proposition implies that Q e,w is combinatorially equivalent to thecube I ℓ ( w ) if X w is of complexity zero, i.e., a reduced decomposition of w consistsof distinct letters. Theorem 5.8. For a permutation w in S n , the following hold:(1) the Schubert variety X w is smooth and of complexity one if and only if Q e,w is combinatorially equivalent to the polytope Perm × I ℓ ( w ) − , and(2) the Schubert variety X w is singular and of complexity one if and only if Q e,w is combinatorially equivalent to the polytope Q e, × I ℓ ( w ) − .Proof. Since both the polytopes Perm × I ℓ ( w ) − and Q e, × I ℓ ( w ) − are ℓ ( w ) − c ( w ) = 1. Hence it suffices to prove the ‘only if’ part in eachstatement.If a Schubert variety X w is smooth and of complexity one, then w has a reduceddecomposition w of the form: w = s i · · · s i k − s i k s i k +1 s i k s i k +2 · · · s i ℓ − and i k +1 = i k + 1 . Hence the polytope Q e,w is combinatorially equivalent to Q e,s ik s ik +1 s ik × I ℓ ( w ) − ∼ =Perm × I ℓ ( w ) − by Lemma 5.1 and Proposition 5.7. If X w is singular and of com-plexity one, then w has a reduced decomposition w of the form: w = s i · · · s i k − s i k +1 s i k s i k +2 s i k +1 s i k +3 · · · s i ℓ − and i k +1 = i k + 1 , i k +2 = i k + 2 . Hence Q e,w is combinatorially equivalent to Q e,s ik +1 s ik s ik +2 s ik +1 × I ℓ ( w ) − ∼ = Q e, × I ℓ ( w ) − by Lemma 5.1 and Proposition 5.7. (cid:3) The first statement in the above theorem shows the equivalence between (1 ′ )and (6 ′ ) in Theorem 1.2, and hence it proves Theorem 1.2. The second statementin the above theorem shows the equivalence between (1 ′′ ) and (6 ′′ ) in Theorem 1.3,and hence it proves Theorem 1.3.Note that for a permutation w in S n , the following hold:(1) w avoids the patterns 3412 and 4231 if and only if w − avoids those patterns,and(2) [321; 3412]( w ) = 1 if and only if [321; 3412]( w − ) = 1.Hence X w is smooth and of complexity one if and only if X w − is smooth and ofcomplexity one. Corollary 5.9. For w ∈ S n , if [321; 3412]( w ) = 1 , then the Bruhat interval poly-topes Q e,w and Q e,w − are combinatorially equivalent. Therefore, for a Schubert variety X w of complexity one, X w is smooth if andonly if the moment polytope Q e,w − is simple. N SCHUBERT VARIETIES OF COMPLEXITY ONE 19 Flag Bott manifolds Like as a Bott–Samelson variety is diffeomorphic to a Bott manifold with ahigher rank torus action, a flag Bott–Samelson variety is diffeomorphic to a flagBott manifold with a higher rank torus action. Whereas a Bott manifold is a toricvariety, a flag Bott manifold is not a toric variety in general, but it becomes a GKMmanifold, see [18]. We have previously observed in Theorem 4.7 that there is anisomorphism between a smooth Schubert variety and a flag Bott–Samelson variety.Using the diffeomorphism between a flag Bott–Samelson variety and a certain flagBott manifold, we will provide a formula for the cohomology ring of a smoothSchubert variety of complexity one.Recall that for a holomorphic vector bundle E over a complex manifold M , thereis an associated flag-bundle F ℓ ( E ) obtained from E by replacing each fiber E p bythe full flag manifold F ℓ ( E p ). Definition 6.1 ([18, Deifnition 2.1]) . A flag Bott tower is an iterated flag-bundle F r π r −→ F r − π r − −→ · · · π −→ F π −→ F = { a point } of manifolds F k = F ℓ (cid:16) C ⊕ L n k m =1 ξ ( m ) k (cid:17) , where ξ ( m ) k is a holomorphic line bundleover F k − for each 1 ≤ m ≤ n k and 1 ≤ k ≤ r . Each F k is called a flag Bottmanifold (of height k ).Because we are considering an iterated full flag-bundle, there is an isomorphism ψ : Z n × · · · × Z n r → Pic( F r ) . Therefore, there is a sequence (cid:16) a ( m )1 ,k , a ( m )2 ,k , . . . , a ( m ) k − ,k (cid:17) of integer vectors with a ( m ) j,k ∈ Z n j which maps to ξ ( m ) k via ψ . Hence a set { a ( m ) j,k } ≤ m ≤ n k , ≤ j Let I = ( I , . . . , I r ) be a sequence of subsetsof [ n ] , where I k = { u k, , . . . , u k,n k } for ≤ k ≤ r . Assume that each I k is aninterval. Then the flag Bott–Samelson variety Z I is diffeomorphic to a flag Bottmanifold F r determined by the vectors a ( m ) j,k = ( a ( m ) j,k (1) , . . . , a ( m ) j,k ( n j )) ∈ Z n j (1 ≤ m ≤ n k and ≤ j < k ≤ r ) where a ( m ) j,k ( p ) = h e u j,p − e u j,nj +1 , e u k,m − e u k,nk +1 i for ≤ p ≤ n j . Here, e u k, , . . . , e u k,nk +1 are the standard basis vectors of R n k +1 for ≤ k ≤ r . Theorem 6.3. Let X w be a smooth Schubert variety of complexity one. Then X w is diffeomorphic to a flag Bott manifold of height ℓ ( w ) − F ℓ ( w ) − F ℓ ( w ) − · · · F F = { a point } , π ℓ ( w ) − π ℓ ( w ) − π π where F q → F q − is a F ℓ ( C ) -bundle for some ≤ q ≤ ℓ ( w ) − and F k → F k − is a C P -bundle for every k = q . Furthermore, the iterated bundle structure iscompletely determined by a reduced decomposition of w . Proof. Let ℓ ( w ) = ℓ . From the proof of Theorem 4.7, we may assume that X w isisomorphic to a flag Bott–Samelson variety Z I , where I = ( I , . . . , I ℓ − ) of length ℓ − I k = { i k } if 1 ≤ k < q, { i, i + 1 } if k = q, { i k +2 } if k > q. Note that w = ( i , . . . , i q − , i, i + 1 , i, i q +3 , . . . , i ℓ ). Thus X w is diffeomorphic to aflag Bott manifold of height ℓ − ≤ k ≤ ℓ − 2, the manifold F k is determined as follows.(1) If 1 ≤ k < q , then n k = 1 and F k = F ℓ ( C ⊕ ξ k ), where ξ k is a holomorphicline bundle determined by a sequence ( a (1)1 ,k , . . . , a (1) k − ,k ) of integers, where a (1) j,k = h e i j − e i j +1 , e i k − e i k +1 i ∈ Z . (2) If k = q , then n j = 1 for j < k = q and n k = 2 and F k = F ℓ ( C ⊕ ξ (1) k ⊕ ξ (2) k ), where ξ (1) k and ξ (2) k determined by sequences ( a (1)1 ,q , . . . , a (1) q − ,q ) and( a (2)1 ,q , . . . , a (2) q − ,q ) of integers, respectively. Here, we have that a (1) j,q = h e i j − e i j +1 , e i − e i +2 i ∈ Z , a (2) j,q = h e i j − e i j +1 , e i +1 − e i +2 i ∈ Z . (3) If k > q , then n k = 1 and F k = F ℓ ( C ⊕ ξ k ), where ξ k is a holomorphicline bundle determined by a sequence ( a (1)1 ,k , . . . , a (1) k − ,k ) of integer vectors,where a (1) j,k = h e i j − e i j +1 , e i k +2 − e i k +2 +1 i if j < q, ( h e i − e i +2 , e i k +2 − e i k +2 +1 i , h e i +1 − e i +2 , e i k +2 − e i k +2 +1 i ) if j = q, h e i j +2 − e i j +2 +1 , e i k +2 − e i k +2 +1 i if j > q. This proves the theorem. (cid:3) Combining Theorem 2.2 with Theorems 4.7 and 6.3, we conclude that everysmooth Schubert variety of complexity ≤ ℓ ( w ) and the complexity c ( w ). Therefore, using [16, Corol-lary 4.4], we provide the cohomology ring H ∗ ( X w ; Z ) of a smooth Schubert variety X w with c ( w ) ≤ Corollary 6.4. Let X w be a smooth Schubert variety of complexity one. Supposethat w = s i . . . s i q − s i s i +1 s i s i q +3 . . . s i ℓ is a reduced decomposition of w . Then thecohomology ring H ∗ ( X w ; Z ) is given as follows: H ∗ ( X w ; Z ) ∼ = Z [ y j, , . . . , y j,n j +1 | ≤ j ≤ ℓ − / h I , . . . , I ℓ − i . N SCHUBERT VARIETIES OF COMPLEXITY ONE 21 Here, y j,k ’s are degree two elements and I k is the ideal given as follows.For k < q : I k = (1 − y k, )(1 − y k, ) − − k − X j =1 a (1) j,k y j, , for k = q : I q = (1 − y q, )(1 − y q, )(1 − y q, ) − − q − X j =1 a (1) j,q y j, − q − X j =1 a (2) j,q y j, , for k > q : I k = (1 − y k, )(1 − y k, ) − − X ≤ j ≤ k − ,j = q a (1) j,k y j, − ( a (1) q,k (1) y q, + a (1) q,k (2) y q, ) . Example 6.5. Suppose that w = s s s s s . Then the Schubert variety X w issmooth of complexity one. By Theorem 6.2, the Schubert variety X w is diffeomor-phic to a flag Bott manifold of height 3 with n = n = 1 and n = 2 which isdetermined by the following integer vectors: a (1)1 , = h e − e , e − e i = − , a (1)1 , = h e − e , e − e i = 0 , a (2)1 , = h e − e , e − e i = 0 , a (1)2 , = h e − e , e − e i = − , a (2)2 , = h e − e , e − e i = 0 . Therefore, by Corollary 6.4, the cohomology ring of X w is H ∗ ( X w ; Z ) ∼ = Z [ y j, , . . . , y y,n j +1 | ≤ j ≤ / h I , . . . , I i , where I = (1 − y , )(1 − y , ) − ,I = (1 − y , )(1 − y , ) − (1 + y , ) ,I = (1 − y , )(1 − y , )(1 − y , ) − (1 + y , ) . This gives relations among generators: y , + y , = y , y , = 0 ,y , + y , + y , = y , y , = 0 ,y , + y , + y , + y , = y , y , + y , y , + y , y , = y , y , y , = 0 . Therefore, by setting y = y , , y = y , , y = y , , y = y , , we have that H ∗ ( X w ; Z ) ∼ = Z [ y , . . . , y ] /I, where I is an ideal generated by y , y ( y + y ) , ( y + y ) y + ( y + y + y ) y , y y ( y + y + y ) . We enclose this section by mentioning other studies on the cohomology ringsof smooth Schubert varieties. Indeed, the cohomology rings of smooth Schubertvarieties are studied in [12] (also, see [9]). We will demonstrate their result fora specific permutation 23541, which is the one considered in Example 6.5. Firstwe recall from [9] that this permutation is related to a partition as follows. For a given partition λ = (0 ≤ λ ≤ · · · ≤ λ n ), one may associate a permutation w = w (1) . . . w ( n ) by the recursive rule w ( i ) = max( { , . . . , λ i } \ { w (1) , . . . , w ( i − } ) . For example, if λ = (2 , , , , X λ the Schubert variety given by the permutation coming from λ . Thepresentation for the cohomology ring of X λ is given as follows:(6.1) H ∗ ( X λ ; Z ) ∼ = Z [ x , . . . , x n ] / h h λ i − i +1 ( i ) | ≤ i ≤ n i where h m ( N ) is the complete homogeneous symmetric function: h m ( N ) := h m ( x , . . . , x N ) = X ≤ i ≤···≤ i m ≤ N x i · · · x i m . For the partition λ = (2 , , , , J := h h λ i − i +1 ( i ) | ≤ i ≤ n i onthe right hand side of (6.1) is generated by the complete homogeneous symmetricpolynomials h (1) = x ,h (2) = x + ( x + x ) x ,h (3) = ( x + x + x ) x + ( x x + x )( x + x ) + ( x + x + x ) x ,h (4) = x + ( x + x ) x + ( x + x + x ) x + ( x + x + x + x ) x ,h (5) = x + x + x + x + x . Moreover, we get that Z [ x , . . . , x ] /J ∼ = Z [ x , . . . , x ] /J ′ , where J ′ = h x , ( x + x ) x , ( x + x + x ) x , ( x + x + x ) x + ( x + x + x + x ) x i . By sending y 7→ − x , y x + x , y x , y x , we obtain an isomorphism Z [ x , . . . , x ] /I ∼ = Z [ y , . . . , y ] /J ′ between the two cohomology ring representa-tions. For instance, we have that( y + y ) y + ( y + y + y ) y ( x + x + x ) x + ( x + x + x + x ) x ,y y ( y + y + y ) x x ( x + x + x + x )= x (( x + x + x ) x + ( x + x + x + x ) x ) − ( x + x + x ) x . Remark 6.6. The notion of flag Bott–Samelson variety can be defined in a gen-eral Lie type, where G is a simply-connected semisimple algebraic group over C .Furthermore, Proposition 4.3, Corollary 4.5, and Theorem 6.2 are still true in ageneral Lie type. Acknowledgements. Lee was supported by IBS-R003-D1. Masuda was sup-ported in part by JSPS Grant-in-Aid for Scientific Research 19K03472 and a bi-lateral program between JSPS and RFBR. Park was supported by the Basic Sci-ence Research Program through the National Research Foundation of Korea (NRF)funded by the Government of Korea (NRF-2018R1A6A3A11047606). This workwas partly supported by Osaka City University Advanced Mathematical Institute(MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JP-MXP0619217849). N SCHUBERT VARIETIES OF COMPLEXITY ONE 23 References [1] The On-Line Encyclopedia of Integer Sequences (OEIS), 2020 (accessed October 9, 2020).[2] Raoul Bott and Hans Samelson, Applications of the theory of Morse to symmetric spaces ,Amer. J. Math. (1958), 964–1029.[3] Michel Brion, Lectures on the geometry of flag varieties , Topics in cohomological studies ofalgebraic varieties, Trends Math., Birkh¨auser, Basel, 2005, pp. 33–85.[4] Michel Brion and S. Senthamarai Kannan, Minimal rational curves on generalized Bott–Samelson varieties , arXiv preprint arXiv:1910.06197 (2019).[5] Fabrizio Caselli, Michele D’Adderio and Mario Marietti, Weak Generalized Lifting Prop-erty, Bruhat Intervals, and Coxeter Matroids , International Mathematics Research Notices,rnaa124, https://doi.ore/10.1093/imrn/rnaa124[6] Daniel Daly, Fibonacci numbers, reduced decompositions, and / pattern classes , Ann.Comb. (2010), no. 1, 53–64.[7] , Reduced decompositions with one repetition and permutation pattern avoidance ,Graphs Combin. (2013), no. 2, 173–185.[8] Michel Demazure, D´esingularisation des vari´et´es de Schubert g´en´eralis´ees , Ann. Sci. ´EcoleNorm. Sup. (4) (1974), 53–88.[9] Mike Develin, Jeremy L. Martin, and Victor Reiner, Classification of Ding’s Schubert vari-eties: finer rook equivalence , Canad. J. Math. (2007), no. 1, 36–62.[10] C. Kenneth Fan, Schubert varieties and short braidedness , Transform. Groups (1998), no. 1,51–56.[11] Naoki Fujita, Eunjeong Lee, and Dong Youp Suh, Algebraic and geometric proper-ties of flag Bott–Samelson varieties and applications to representations , arXiv preprintarXiv:1805.01664v3, to appear in Pacific J. Math.[12] Vesselin N. Gasharov and Victor Reiner, Cohomology of smooth Schubert varieties in partialflag manifolds , J. London Math. Soc. (2) (2002), no. 3, 550–562.[13] Michael Grossberg and Yael Karshon, Bott towers, complete integrability, and the extendedcharacter of representations , Duke Math. J. (1994), no. 1, 23–58.[14] Robin Hartshorne, Algebraic geometry , Springer-Verlag, New York-Heidelberg, 1977, Gradu-ate Texts in Mathematics, No. 52.[15] Jens Carsten Jantzen, Representations of algebraic groups , second ed., Mathematical Surveysand Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003.[16] Shizuo Kaji, Shintarˆo Kuroki, Eunjeong Lee, and Dong Youp Suh, Flag bott manifolds ofgeneral Lie type and their equivariant cohomology rings , Homology Homotopy Appl. (2020), no. 1, 375–390.[17] Paramasamy Karuppuchamy, On Schubert varieties , Comm. Algebra (2013), no. 4, 1365–1368.[18] Shintarˆo Kuroki, Eunjeong Lee, Jongbaek Song, and Dong Youp Suh, Flag Bott manifoldsand the toric closure of a generic orbit associated to a generalized Bott manifold , arXivpreprint arXiv:1708.02082v3, to appear in Pacific J. Math.[19] Venkatramani Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varietiesin Sl( n ) /B , Proc. Indian Acad. Sci. Math. Sci. (1990), no. 1, 45–52.[20] Eunjeong Lee and Mikiya Masuda, Generic torus orbit closures in Schubert varieties , J.Combin. Theory Ser. A (2020), 105143, 44.[21] Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, Toric Bruhat interval polytopes , arXivpreprint arXiv:1904.10187.[22] Nicolas Perrin, Small resolutions of minuscule Schubert varieties , Compos. Math. (2007),no. 5, 1255–1312.[23] Bridget Eileen Tenner, Pattern avoidance and the Bruhat order , J. Combin. Theory Ser. A (2007), no. 5, 888–905.[24] , Repetition in reduced decompositions , Adv. in Appl. Math. (2012), no. 1, 1–14.[25] Emmanuel Tsukerman and Lauren K. Williams, Bruhat interval polytopes , Adv. Math. (2015), 766–810. (E. Lee) Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang37673, Republic of Korea Email address : [email protected] (M. Masuda) Osaka City University Advanced Mathematical Institute & Departmentof Mathematics, Graduate School of Science, Osaka City University, Sumiyoshi-ku,Sugimoto, 558-8585, Osaka, Japan Email address : [email protected] (S. Park) Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea Email address ::