0-0 Ten Fantastic Facts on Bruhat Order Sara Billey - PDF document

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Ten Fantastic Facts on Bruhat Order Sara Billey http://www.math.washington.edu/ ∼ billey/classes/581/bulletins/bruhat.ps 0-1

Bruhat Order on Coxeter Groups generators : s 1 , s 2 , . . . s n Coxeter Groups. i = 1 and ( s i s j ) m ( i,j ) = 1 s 2 relations : Coxeter Graph. V = { 1 , . . . , n } , E = { ( i, j ) : m ( i, j ) ≥ 3 } . Define. If w ∈ W = Coxeter Group, • w = s i 1 s i 2 . . . s i p is a reduced expression if p is minimal. • l ( w ) = length of w =p. Example. S n = Permutations generated by s i = ( i ↔ i +1) , i < n , with relations s i s i = 1 ( s i s j ) 2 = 1 if | i − j | > 1 ( s i s i +1 ) 3 = 1 w = 4213 = s 1 s 3 s 2 s 1 and l ( w ) = 4 Other Examples. Weyl groups and dihedral groups. 0-2

Bruhat Order on Coxeter Groups Natural Partial Order on W. v ≤ w if any reduced expression for w contains a subexpression which is a reduced expression for v . Example. s 1 s 3 s 2 s 1 > s 3 s 1 > s 1 0-3

Chevalley-Bruhat Order on Coxeter Groups Natural Partial Order on W. v ≤ w if any reduced expression for w contains a subexpression which is a reduced expression for v . Example. s 1 s 3 s 2 s 1 > s 3 s 1 > s 1 0-4

Ehresmann-Chevalley-Bruhat Order on Cox- eter Groups Natural Partial Order on W. v ≤ w if any reduced expression for w contains a subexpression which is a reduced expression for v . Example. s 1 s 3 s 2 s 1 > s 3 s 1 > s 1 0-5

Bruhat-et.al Order on Coxeter Groups Natural Partial Order on W. v ≤ w if any reduced expression for w contains a subexpression which is a reduced expression for v . Example. s 1 s 3 s 2 s 1 > s 3 s 1 > s 1 0-6

Bruhat-et.al Order on Coxeter Groups • v ≤ w if any reduced expression for w contains a subexpression which is a reduced expression for v . • v ≤ w if every reduced expression for w contains a subexpression which is a reduced expression for v . • Covering relations: w covers v ⇐ ⇒ w = s i 1 s i 2 . . . s i p (reduced) and there exists j such that v = s i 1 . . . � s i j . . . s i p (reduced). • Covering relations: w covers v ⇐ ⇒ w = vt and l ( w ) = l ( v ) + 1 where t ∈ { us i u − 1 : u ∈ W } = Reflections in W . 0-7

Bruhat-et.al Order on Coxeter Groups t t t ✟✟ ❍ t ❍ ✘✘✘✘ t t ❳ ❳ ❳ t t ❳ ❳ ❳ ❳ t t ❳ ✟✟ ❍ ✟✟ ❍ t t ❍ ❍ ❍ ✟✟ ❍ ✟✟ t t t ❍ ❍ ✘✘✘✘ t t ✘✘✘✘ t t ❳ ❳ ❳ t t ❳ ❍ ✟✟ t t ❍ t t t t Quotient E 6 modulo S 6 0-8

Fact 1: Bruhat Order Characterizes Inclusions of Schubert Varieties � • Bruhat Decomposition : G = GL n = BwB w ∈ S n • Flag Manifold : G/B a complex projective smooth variety for any semisimple or Kac-Moody group G and Borel subgroup B • Schubert Cells : BwB/B • Schubert Varieties : BwB/B = X ( w ) Chevalley. (ca. 1958) X ( v ) ⊂ X ( w ) if and only if v ≤ w i. e. � BwB/B = BvB/B v ≤ w � e polynomial for H ∗ ( X ( w )) is P w ( t 2 ) = t 2 l ( v ) = ⇒ . The Poincar´ v ≤ w 0-9

Fact 2: Contains Young’s Lattice • Grassmannian Manifold : { k -dimensional subspaces of C n } = GL n /P for P =maximal parabolic subgroup. • Schubert Cells : BwB/P indexed by elements of W J = W/ � s i : i ∈ J � � • Schubert Varieties : X ( w ) = BwB/P = BvB/P . w ≥ v ∈ W J • Elements of W J can be identified with partitions inside a box, and the induced order is equivalent to containment of partitions. 0-10

Fact 3: Nicest Possible M¨ obius Function obius Function on a Poset: unique function µ : { x < y } → Z such that M¨ � � 1 x = z µ ( x, y ) = 0 x � = z. x ≤ y ≤ z Theorem. (Verma, 1971) µ ( x, y ) = ( − 1) l ( y ) − l ( x ) if x ≤ y . � [ x, y ] J = [ x, y ] ( − 1) l ( y ) − l ( x ) Theorem. (Deodhar, 1977) µ ( x, y ) J = . 0 otherwise Apply M¨ obius Inversion to • Kazhdan-Lusztig polynomials. • Kostant polynomials • Any family of polynomials depending on Bruhat order. 0-11

Fact 4: Beautiful Rank Generating Functions � � t l ( u ) = a k t k rank generating function: W ( t ) = u ∈ W k ≥ 0 Computing W ( t ). for W = finite reflection group � (1 + t + t 2 + · · · + t e i ) • W ( t ) = (Chevalley) t ht ( α )+1 − 1 � • W ( t ) = (Kostant ’59, Macdonald ’72) t ht ( α ) − 1 α ∈ R + Here, e ′ i s = exponents of W , R + =positive roots associated to W and s 1 , . . . , s n , ht ( α ) = k if α = α i 1 + · · · + α i k (simple roots). 0-12

Fact 4: Beautiful Rank Generating Functions • Carrell-Peterson, 1994 : If X ( w ) is smooth t ht ( β )+1 − 1 � � t l ( v ) = P [ˆ 0 ,w ] ( t ) = t ht ( β ) − 1 v ≤ w β ∈ R + σ β ≤ w • Gasharov : For w ∈ S n , if X ( w ) is rationally smooth � (1 + t + t 2 + · · · + t d i ) P [ˆ 0 ,w ] ( t ) = for some set of d i ’s. • In 2001, Billey and Postnikov gave similar factorizations for all ratio- nally smooth Schubert varieties of semisimple Lie groups. 0-13

Fact 5: Symmetric Interval [ˆ 0 , w ] = ⇒ X ( w ) is Rationally Smooth Definition. A variety X of dimension d is rationally smooth if for all x ∈ X , � 0 i � = 2 d H i ( X, X \ { x } , Q) = i = 2 d. Q Theorem. (Kazhdan-Lusztig ’79) X ( w ) is rationally smooth if and only if the Kazhdan-Lusztig polynomials P v,w = 1 for all v ≤ w . Theorem. (Carrell-Peterson ’94) X w is rationally smooth if and only if [ˆ 0 , w ] is rank symmetric. 0-14

Fact 5: Symmetric Interval [ˆ 0 , w ] = ⇒ X ( w ) is Rationally Smooth 0-15

Fact 6: [ x, y ] Determines the Composition Series for Verma Modules • g = complex semisimple Lie algebra • h = Cartan subalgebra • λ = integral weight in h ∗ • M ( λ ) = Verma module with highest weight λ • L ( λ ) = unique irreducible quotient of M ( λ ) • W = Weyl group corresponding to g and h Fact. { L ( λ ) } λ ∈ h ∗ = complete set of irreducible highest weight modules. Problem. Determine the formal character of M ( λ ) � ch( M ( λ )) = [ M ( λ ) : L ( µ )] · ch( L ( µ )) µ 0-16

Fact 6: [ x, y ] Determines the Composition Series for Verma Modules Answer. Only depends on Bruhat order using the following reasoning:   λ = x · λ 0  • [ M ( λ ) : L ( µ )] � = 0 ⇐ ⇒ µ = y · λ 0   x < y ∈ W (Verma, Bernstein-Gelfand-Gelfand, van den Hombergh) • [ M ( x · λ 0 ) : L ( y · λ 0 )] = m ( x, y ) independent of λ 0 . (BGG ’75) ⇒ # { r ∈ R : x < rx ≤ z } = l ( z ) − l ( x ) • m ( x, y ) = 1 ⇐ ∀ x ≤ z ≤ y. (Janzten ’79) m ( x, y ) = P x,y (1) = Kazhdan-Lusztig polynomial for x < y (Beilinson-Bernstein ’81, Brylinski-Kashiwara ’81) 0-17

Fact 6: [ x, y ] Determines the Composition Series for Verma Modules Conjecture. The Kazhdan-Lusztig polynomial P x,y ( q ) depends only on the interval [ x, y ] (not on W or g etc. ) Example. y s � ❅ ❅ � s s ❅ � ❅ � � ❅ = ⇒ m ( x, y ) = 1 s s ❅ ❅ � � s x 0-18

Fact 7: Order Complex of ( u, v ) is Shellable • Order complex ∆( u, v ) has faces determined by the chains of the open interval ( u, v ) , maximal chains determine the facets. • ∆ = pure d -dim complex is shellable if the maximal faces can be linearly ordered C 1 , C 2 , . . . such that for each k ≥ 1 , ( C 1 ∪ · · · ∪ C k ) ∩ C k +1 is pure ( d − 1) -dimensional. Shellable Not Shellable 0-19

Fact 7: Order Complex of ( u, v ) is Shellable Lexicographic Shelling of [ u, v ] : (Bjorner-Wachs ’82, Proctor, Edelman) • Each maximal chain → label sequence v = s 1 s 2 . . . s p > s 1 . . . � s j . . . s p > s 1 . . . � s i . . . � s j . . . s p > . . . maps to ( j, i, . . . ) • Order chains by lexicographically ordering label sequences. Consequences: 1. ∆( u, w ) J is Cohen-Macaulay. � ( u, w ) J = ( u, w ) the sphere S l ( w ) − l ( u ) − 2 2. ∆( u, w ) J ≡ the ball B l ( w ) − l ( u ) − 2 otherwise 0-20

Fact 8: Rank Symmetric, Rank Unimodal and k -Sperner 1. P = ranked poset with maximum rank m 2. P is rank symmetric if the number of elements of rank i equals the number of elements of rank m − i . 3. P is rank unimodal if the number of elements on each rank forms a unimodal sequence. 4. P is k -Sperner if the largest subset containing no ( k + 1) -element chain has cardinality equal to the sum of the k middle ranks. Theorem. (Stanley ’80) For any subset J ⊂ { s 1 , . . . s n } , let W J be the partially ordered set on the quotient W/W J induced from Bruhat order. Then W J is rank symmetric, rank unimodal, and k -Sperner. (proof uses the Hard Lefschetz Theorem) 0-21

Fact 9: Efficient Methods for Comparison Problem. Given two elements u, v ∈ W , what is the best way to test if u < w ? Don’t use subsequences of reduced words if at all possible. Tableaux Comparison in S n . (Ehresmann) • Take u = 352641 and v = 652431 . • Compare the sorted arrays of { u 1 , . . . u i } ≤ { v 1 , . . . , v i } : 3 ≤ 6 3 5 ≤ 5 6 2 3 5 ≤ 2 5 6 2 3 5 6 ≤ 2 4 5 6 2 3 4 5 6 ≤ 2 3 4 5 6 1 2 3 4 5 6 ≤ 1 2 3 4 5 6 0-22

Fact 9: Efficient Methods for Comparison • Generalized to B n and D n and other quotients by Proctor (1982). • Open: Find an efficient way to compare elements in E 6 , 7 , 8 in Bruhat order. Another criterion for Bruhat order on W . u ≤ v in W J for each maximal proper J ⊂ u ≤ v in W ⇐ ⇒ { s 1 , s 2 , . . . , s n } . 0-23