Enumeration of Flags in Eulerian Posets

Transcription

1 Enumeration of Flags in Eulerian Posets Louis Billera Cornell University, Ithaca, NY * * * * * Conference on Algebraic and Geometric Combinatorics Anogia, Crete August, 2005

2 Lecture I: Faces in polytopes and chains in posets Lecture II: Enumeration algebra, quasisymmetric functions and the peak algebra Lecture III: Applications: Coxeter groups arrangements, convex closures and

3 Lecture I: Faces in polytopes and chains in posets f-vectors of convex polytopes and the g-theorem flag f-vectors of graded posets Eulerian posets and the cd-index Convolutions of flag f-vectors 1

4 Preamble on f-vectors of polytopes For a d-dimensional convex polytope Q, let f i = f i (Q) = the number of i-dimensional faces of Q f 0 = the number of vertices, f 1 = the number of edges,. f d 1 = the number of facets (or defining inequalities) The f-vector of Q f(q) = (f 0, f 1,..., f d 1 ) Problem: Determine when a vector f = (f 0, f 1,..., f d 1 ) is f(q) for some d-polytope Q. d = 2: Exercise d = 3: Steinitz (1906) d 4: open 2

5 Simplicial polytopes A polytope is simplicial if all faces are simplices (equiv: vertices are in general position) The h-vector (h 0,..., h d ) of a simplicial d-polytope is defined by the polynomial relation d i=0 h i x d i = d i=0 f i 1 (x 1) d i. The corresponding g-vector (g 0,..., g d/2 ) is defined by g 0 = 1 and g i = h i h i 1, for i 1. The h-vector and the f-vector of a polytope mutually determine each other via the formulas (for 0 i d): h i = i j=0 ( 1) i j( d j i j ) fj 1, f i 1 = i j=0 ( d j i j ) hj, 3

6 The g-theorem Theorem(BL/S,1980): (h 0, h 1,..., h d ) is the h-vector of a simplicial convex d-polytope if and only if h i = h d i (Dehn-Sommerville equations) for all i, and g i = h i h i 1, 0 i d 2, satisfy and for i 1. g i 0 (Generalized Lower Bound Thm) g i+1 g i i (Macaulay-McMullen conditions) Note: The last conditions derive from (but are not quite the same as) the conditions of Kruskal and Katona for f-vectors of general simplicial complexes, but with g i in place of f i. Equivalently, the g i s are a Hilbert function. Necessity proof (Stanley) depends on producing a commutative ring with this Hilbert function. 4

7 General polytopes For general convex polytopes, the situation for f-vectors is much less satisfactory. 1) The only equation they all satisfy is the Euler relation f 0 f 1 + f 2 ± f d 1 = 1 ( 1) d 2) Already in d = 4, we do not know all linear inequalities on f-vectors. 3) There is little hope at this point of giving an analog to the Macaulay-McMullen conditions. A possible solution is to try to solve a harder problem: not faces, but chains of faces. count 5

8 Flag f-vectors of Polytopes (first pass) For a d-dimensional polytope Q and a set S of possible dimensions, define f S (Q) to be the number of chains of faces of Q having dimensions prescribed by the set S. The function is called the flag f-vector of Q. S f S (Q) It includes the f-vector, by counting chains of one element: (f S : S = 1). It has a straightforwardly defined flag h-vector that turns out to be a (finely graded) Hilbert function. It satisfies an analog of the Dehn-Sommerville equations, which cut their dimension down to the Fibonacci numbers (compared to n 2 ). And more... 6

9 Face Lattices of Polytopes The best setting in which to study the flag f-vector or a d- polytope Q is that of its lattice of faces P = F(Q), a graded poset of rank d + 1 Q a d e b c P = F(Q) ab. abcd abe bce cde ade bc Q cd ad ae be ce de. a b c d e 7

10 Flag f-vectors of Graded Posets P a graded poset (with 0 and 1) of rank n + 1, with rank function ρ : P N. Flag f-vector is the function S f S = f S (P ), where for S = {i 1,..., i k } [n] := {1,..., n}, f S = #{y 1 < y 2 < < y k y j P, ρ(y j ) = i j } To begin to understand flag f-vectors of convex polytopes, it might be helpful to first be able to answer: Question 1: Determine all flag f-vectors of graded posets, Question 1a: Determine all linear inequalities satisfied by flag f-vectors of graded posets. The former is a Kruskal-Katona analog and remains open. The latter are DS and GLB analogs for graded posets. They have complete solutions (B & Hetyei, 1998). 8

11 Eulerian Posets P is Eulerian if for all x < y P, µ(x, y) = ( 1) ρ(y) ρ(x) where µ is the Möbius function of P. (Equivalently, number of elements of even rank in [x, y] = number of elements of odd rank.) Face posets of polytopes and spheres are Eulerian. Again, two natural questions arise: Question 2: Determine all flag f-vectors of Eulerian posets, Question 2a: Determine all linear inequalities satisfied by flag f-vectors of Eulerian posets. The linear equations are known: For Eulerian posets, only Fibonacci many f S are linearly independent. 9

12 Generalized Dehn-Sommerville Equations There are 2 n flag numbers f S, S [n] for graded posets of rank n + 1. For Eulerian posets, these are not independent: n= 0: f n= 1: f, f {1} but f {1} = 2f (Euler relation) n= 2: f, f {1}, f {2}, f {1,2} but f {1} = f {2} (Euler relation) and f {1,2} = 2f {2} n= 3: f, f {1}, f {2}, f {3}, f {1,2},..., f {1,2,3} but f {1} f {2} + f {3} = 2f (Euler relation), f {1,2} = 2f {2}, etc. n=4: f, f {1}, f {2}, f {3}, f {1,3} The relevant relations for P are all derived from Euler relations in P and in intervals [x, y] of P. Details later... 10

13 The cd-index for Eulerian posets For S [n] let the flag h-vector be defined by h S = T S ( 1) S T f T and for noncommuting indeterminates a and b let u S = u 1 u 2 u n, where u i = b if i S a if i / S Let c = a + b and d = ab + ba. Then for Eulerian posets, the generating function Ψ P = S h S (P )u S is always a polynomial in c and d; this polynomial Φ P (c, d) is called the cd-index of P. 11

14 cd monomials and cd coefficients If P has rank n + 1 then the degree of Φ P (c, d) is n. There are Fibonacci many cd monomials of degree n. Write Φ P = w [w] P w over cd-words w. Stanley: [w] P 0 for polytopes (S-shellable CW spheres) Karu: [w] P 0 for all Gorenstein posets. B, Ehrenborg & Readdy: Among all n-dimensional zonotopes, the cd-index is termwise minimized on the n-cube C n. B & Ehrenborg: Among all n-dimensional polytopes, the cdindex is termwise minimized on the n-simplex n. 12

15 An example: The Boolean algebra B 3 Ex. For P = B 3 = 2 [3], {a, b, c} {b} {a, c} {a, b} {b, c} {a} {c} = faces of a c b f = 1, f {1} = 3, f {2} = 3, f {1,2} = 6 so h = 1, h {1} = 2, h {2} = 2, h {1,2} = 1, and so Ψ P = aa + 2ba + 2ab + bb = (a + b) 2 + (ab + ba) = c 2 + d = Φ P 13

16 Convolutions of flag f-vectors Notation: Write f (n) S, S [n 1], when counting chains in a poset of rank n. Given f (n) S and f (m) T, S [n 1], T [m 1] and P a poset of rank n + m, define f (n) S f (m) T (P ) = x P : r(x)=n f (n) S ([ˆ0, x]) f (m) T ([x, ˆ1]) Claim: f (n) S f (m) T = f (n+m) S {n} (T +n) where T + n := {x + n x T } 14

17 If rank P is n + m f (n) S f (m) T (P ) = x P : r(x)=n f (n) S ([ˆ0, x]) f (m) T ([x, ˆ1]) P 1 x [x,1] [0,x] 0 f (n) S f (m) T = f (n+m) S {n} (T +n) Ex. f (2) {1} f (3) {2} = f (5) {1,2,4} f (2) f (3) = f (5) {2} 15

18 Convolved Inequalities Kalai: If F = α S f (n) S, G = β S f (m) S where F (P 1 ) 0 and G(P 2 ) 0 for all polytopes (graded posets, Eulerian posets) P 1 and P 2 of ranks n and m, respectively, then F G(P ) 0 for all polytopes (graded posets, Eulerian posets) P of rank n+m Ex. Polygons have at least 3 vertices, so f (3) {1} 3f (3) 0 for all polygons (rank = dimension + 1). Thus ( f (3) {1} 3f (3) ) f (1) = f (4) {1,3} 3f (4) {3} 0 for all 3-polytopes (number of vertices in 2-faces 3 times the number of 2-faces). 16

19 Derived Inequalities for Polytopes Most of the inequalities described earlier are of the form F (P ) = α S f (n) S (P ) 0 and so can be convolved to give further inequalities. For example: Let w = c n 1dc n 2dc n 3 c n pdc n p+1 be a cd-word, and define m 0,..., m p by m 0 = 1 and m i = m i 1 + n i + 2. Then the coefficient of w in the cd-index is given by i 1,...,i p ( 1) (m 1 i 1 )+(m 2 i 2 )+ +(m p i p ) k i1 i 2 i p where the sum is over all p-tuples (i 1, i 2,..., i p ) such that m j 1 i j m j 2 and k S = ( 2) S T f T T S 17

20 cd coefficients as forms The cd-indices for posets of ranks 1 5: f (1) f (2) c f (3) ( c 2 + f (3) {1} f (4) ( c 3 + f (4) {1} f (5) ( c f (5) {1} ( f (5) ( + ) (3) 2f d ) ( (4) 2f dc+ f (4) {2} f (4) ) cd {1} ( (5) 2f )dc 2 f (5) {3} f (5) {2} + f (5) {1} f (5) {1,3} 2f (5) {3} {2} f (5) {1} 2f (5) 2f (5) {1} ) cdc ) c 2 d + 4f (5) ) d 2 18

21 Relations on the f S Polytopes of dimension d 1 (Eulerian posets of rank d) satisfy the Euler relations: f (d) f (d) {1} + f (d) {2} + ( 1) d 1 f (d) {d 1} + ( 1)d f (d) = 0 Thm (Bayer & B.): All linear relations on the f (d) S for polytopes, and so for Eulerian posets, come from these via convolution. Proof consists of producing Fibonacci many polytopes whose flag f-vectors span. These can be made by considering repeated pyramids (P ) and prisms (B) starting with an edge, never taking two B s in a row. (Count the number of words of length d 1 in P and B with no repeated B, get Fibonacci number.) 19

22 Lecture I: Faces in polytopes and chains in posets Lecture II: Enumeration algebra, quasisymmetric functions and the peak algebra Lecture III: Applications: Coxeter groups arrangements, convex closures and

23 Lecture II: Enumeration algebra, quasisymmetric functions and the peak algebra The enumeration algebra over Eulerian posets Quasisymmetric functions and P -partitions The quasisymmetric function of a graded poset Peak quasisymetric functions and enriched P -partitions Connection to the enumeration algebra via Hopf algebra duality 20

24 Subsets Compositions Let [n] := {1,..., n}. Then β = (β 1,..., β k ) = n + 1 means each β i > 0, and β β k = n + 1. β = (β 1,..., β k ) = n + 1 S(β) := {β 1, β 1 + β 2,..., β β k 1 } [n] and S = {i 1, i 2,..., i k 1 } [n] β(s) := (i 1, i 2 i 1, i 3 i 2,..., n + 1 i k 1 ) = n

25 Enumeration algebras Let A = Q y 1, y 2,... = A 0 A 1 A 2 be the free associative algebra on noncommuting y i, deg(y i ) = i. Via the association y β := y β1 y βk β = (β 1,..., β k ) = n + 1 f S(β) = f S S [n] multiplication in A is the analogue of Kalai s convolution of flag f-vectors, in which f (n) S f (m) T = f (n+m) S {n} (T +n) This corresponds to summing over faces or links of a fixed rank. Ex. f (3) {1} = y 1 y 2 so f (3) {1} f (3) {1} = y 1 y 2 y 1 y 2 = f (6) {1,3,4} 22

26 Euler elements of A n+1 F A n functionals on graded posets of rank n, i.e., expressions of the form S [n 1] α Sf (n) S. Ex. As an element of A 4 2y 4 y 1 y 3 + y 2 y 2 y 3 y 1 = the Euler relation for posets of rank 4. 2f (4) f (4) {1} + f (4) {2} f (4) {3} For k 1 define in A k χ k := i+j=k ( 1) i y i y j = k i=0 ( 1) i f (k) i, the k th Euler relation, where y 0 = 1 and f (k) 0 = f (k) k = f (k). 23

27 Eulerian Enumeration Algebra Ex. In A 4 χ 4 = y 0 y 4 y 1 y 3 + y 2 y 2 y 3 y 1 + y 4 y 0 Let I E = χ k : k 1 A 2-sided ideal of all relations on Eulerian posets A E = A/I E algebra of functionals on Eulerian posets. Theorem (B. & Liu): As graded algebras, A E = Q y1, y 3, y 5,... ( odd jump algebra), and so dim Q (A E ) n is the n th Fibonacci number. 24

28 Quasisymmetric functions Let Q Q[[x 1, x 2,... ]] the algebra of quasisymmetric functions Q := Q 0 Q 1 where Q n := span{m β β = (β 1,..., β k ) = n} M β := i 1 <i 2 < <i k x β 1 i 1 x β 2 i 2 x β k i k. Here M 0 = 1 so Q 0 = Q; otherwise k > 0, each β i > 0, and β β k = n. Ex. (1, 2, 1) = 4 and M (1,2,1) = i 1 <i 2 <i 3 x 1 i 1 x 2 i 2 x 1 i 3 25

29 Relabel M β : For S [n], define M S = M (n+1) S := M β(s) Ex. If S = {1, 3} [3] then β = (1, 2, 1) = 4 and so M {1,3} = M (4) {1,3} = M (1,2,1) Note: Quasisymmetric functions arise naturally as weight enumerators of P -partitions of labeled posets (Gessel). In this context, a more natural basis arises as weight enumerators of labeled chains: L S = M T T S Here S T [n] and S is the descent set of the labeling. 26

30 P -partitions P = {1 < 2 < 3 < } positive integers, P an arbitrary poset and γ : P [n] a 1-1 labeling of P, where n = P. A P -partition is an order preserving function that is nearly strict, i.e., p < q f : P P f(p) < f(q) or f(p) = f(q) and γ(p) < γ(q) Ex. 1) P = n-element chain (naturally labeled): P -partitions are partitions f(1) f(2) f(n) of f(1) + f(2) + + f(n). Ex. 2) P = n-element antichain: P -partitions are compositions (f(1), f(2),..., f(n)) of f(1) + f(2) + + f(n). 27

31 Weight Enumerators Weight enumerator of all P -partitions Γ(P, γ) = f P partition p P x f(p) Proposition(Gessel): 1. If P = n, then Γ(P, γ) Q n. 2. For P a chain labeled γ(1), γ(2),..., γ(n), Γ(P, γ) depends only on the descent set of γ, in fact, where Γ(P, γ) = L S, S = {i γ(i) > γ(i + 1)}. 28

32 Shuffle Product on Q Poset sum P + Q: x P +Q y x, y P, x P y or x, y Q, x Q y Proposition: Γ ( (P, γ) + (Q, δ) ) = Γ(P, γ) Γ(Q, δ) and so L S L T = R L R, where the sum is over all descent sets of shuffles of a sequence with descent set S with one having descent set T. Ex. L (2) {1} L(2) {1} = L(4) {1,2,3} + 2L(4) {1,3} + L (4) {1,2} + L(4) {2} + L(4) {2,3} 29

33 Shuffle Example: L (2) {1} L(2) {1} Both 21 and 43 have descent set {1}, so consider the six shuffles of these two sequences and their descent sets: {1, 3} {2} {2, 3} {1, 3} {1, 2} {1, 2, 3} Thus L (2) {1} L(2) {1} = L(4) {1,2,3} + 2L(4) {1,3} + L (4) {1,2} + L(4) {2} + L(4) {2,3} 30

34 Connection to flag f-vectors Given graded poset P with rank function ρ( ), we associate the formal quasisymmetric function F (P ) = ˆ0=t 0 t 1 t k =ˆ1 x ρ(t 0,t 1 ) 1 x ρ(t 1,t 2 ) 2 x ρ(t k 1,t k ) k with the sum over all multichains in P and ρ(x, y) = ρ(y) ρ(x). F (P ) Q ρ(p ) Ehrenborg: This association is multiplicative, in the sense that F (P 1 P 2 ) = F (P 1 ) F (P 2 ). In this context, changing invariants for P corresponds to changing basis in Q: F (P ) = S f S (P )M S = S h S (P )L S 31

35 The algebra of Peak Functions For a cd-word w of degree n, w = c n 1dc n 2d c n kdc m (deg c = 1, deg d = 2), let I w = {{i 1 1, i 1 }, {i 2 1, i 2 },..., {i k 1, i k }}, where i j = deg(c n 1dc n 2d c n jd). b[i w ] = {S [n] S I, I I w } The peak algebra Π is defined to be the subalgebra of Q generated by the peak quasisymmetric functions Θ w = T b[i w ] 2 T +1 M T, where w is any cd-word (including empty cd-word 1, for which I 1 = ). Fibonacci many! 32

36 Why Π? Stembridge: Peak quasisymmetric functions arise naturally as weight enumerators of enriched P -partitions of labeled posets, where we associate w = c n 1dc n 2d c n kdc m a cd-word of degree n (deg c = 1, deg d = 2) S w = {i 1, i 2,..., i k } [n] where i j = deg(c n 1dc n 2d c n jd). Stembridge considers the basis for Π to be indexed by sets S of the form S w. In this context, his basis Θ S arises as weight enumerators of labeled chains, where S is the peak set of the labeling. (A peak is a descent preceded by an ascent.) 33

37 Enriched P -partitions Z = { 1 < 1 < 2 < 2 < 3 < 3 < } nonzero integers, P an arbitrary poset and γ : P [n] a 1-1 labeling of P, where n = P. An enriched P -partition is an order preserving function that is nearly strict, i.e., p < q f : P Z f(p) < f(q) or f(p) = f(q) > 0 and γ(p) < γ(q) f(p) = f(q) < 0 and γ(p) > γ(q) Weight enumerator of all enriched P -partitions (P, γ) = f enriched P partition x f(p) p P 34

38 Peak Sets Proposition(Stembridge): If P = n, then (P, γ) Π n. For P a chain labeled γ(1), γ(2),..., γ(n), (P, γ) depends only on the peak set of γ, in fact, where (P, γ) = Θ S, S = {i γ(i 1) < γ(i) > γ(i + 1)}. If Π n = Π Q n, then dim Q (Π n ) = a n, the n th Fibonacci number (indexed so a 1 = a 2 = 1). Multiplication of the Θ S has a shuffle interpretation, but this time in terms of peaks. 35

39 Brief interlude on dual Hopf algebras The product on an algebra A can be viewed as a linear map A A A, a b a b A coalgebra C has instead a coproduct C C C A Hopf algebra H has both (and more). In the dual vector space H to a Hopf algebra H, the adjoint of the product on H H H H gives a coproduct on H, and the adjoint of the coproduct on H H H H gives a product on H, making H a Hopf algebra as well. 36

40 Coproducts on A and Q Π and A E both have Fibonacci Hilbert series; not isomorphic (Π commutative, A E not). Bergeron, Mykytiuk, Sottile, van Willigenburg: coproducts on Q and A (M β ) = (y k ) = β=β 1 β 2 M β1 M β2 i+j=k y i y j extend to coproducts on Π and A E, resp. Ex. ( ) M (2,1,1) = 1 M(2,1,1) + M (2) M (1,1) + M (2,1) M (1) + M (2,1,1) 1 (y 2 ) = 1 y 2 + y 1 y 1 + y

41 Hopf Duality Theorem (BMSV): These coproducts make Π and A E into a dual pair of Hopf algebras. For graded poset P, recall the formal quasisymmetric function F (P ) = ˆ0=t 0 t 1 t k =ˆ1 x ρ(t 0,t 1 ) 1 x ρ(t 1,t 2 ) 2 x ρ(t k 1,t k ) k = S f S (P )M S Corollary: If P is Eulerian, then F (P ) Π. Question: How to represent F (P ) in terms of the basis of peak functions {Θ w } for Π? Equivalently, what is the dual basis in A E to the basis {Θ w }? 38

42 Theorem(B.,Hsiao & van Willigenburg): If P is any Eulerian poset, then F (P ) = w 1 2 w d +1 [w] P Θ w, where the [w] P are the coefficients of the cd-index of P and w d is the number of d s in w. Corollary: The elements 1 2 w d +1 [w] A E form a dual basis to the basis Θ w in Π. As a consequence of a result of B., Ehrenborg and Readdy, we get a slick way to see the relationship between enumerative invariants of hyperplane arrangements and zonotopes and those of the associated geometric lattices. 39

43 Lecture I: Faces in polytopes and chains in posets Lecture II: Enumeration algebra, quasisymmetric functions and the peak algebra Lecture III: Applications: Coxeter groups arrangements, convex closures and

44 Lecture III: Applications: Coxeter groups arrangements, convex closures and The Stembridge map Q Π Geometric lattices and arrangements Meet distributive lattices and convex closures Kazhdan-Lusztig polynomials of Coxeter groups 40

45 From Descents to Peaks Ascents Descents Peaks 41

46 The Stembridge map ϑ Consider the (algebra) map ϑ : Q Π defined by associating the weight enumerator of P -partitions to that of enriched P -partitions for the same labeled poset (P, γ); considering chains, we get where for S [n], ϑ(l S ) = Θ Λ(S), Λ(S) = {i S i 1, i 1 / S}. Note that if S is a descent set then Λ(S) is the associated peak set. Proposition: If poset P has a nonnegative flag h-vector (say, if P is Cohen-Macaulay), then ϑ(f (P )) has a nonnegative cdindex. 42

47 Arrangements of hyperplanes Six planes in general position in R 3 A B C D The number of i-gons in each: i A B C D All different, yet each has 32 regions (in fact, each has same flag f-vector). 43

48 The braid arrangement for S Planes: ( ) 4 2 = 6 of the form xi = x j (i < j) Regions: sortings of the coordinates and so correspond to one of the 4! = 24 permutations 44

49 The cd-index of zonotopes and arrangements Let z 1, z 2,..., z m R n and let L be the geometric lattice of subspaces spanned by subsets of the z i s, ordered by inclusion. L is graded, and so is L 0 (add a new 0 to L, increasing the rank by one). Thus F (L 0) Q. Now consider the arrangement H of m hyperplanes {H 1, H 2,..., H m } in R n having normals z 1, z 2,..., z m Note that L can be seen as the lattice of all intersections of the hyperplanes H i, ordered by reverse inclusion (the intersection lattice ). H carves the (n 1)-sphere in R n into regions, that can be ordered by inclusion (and so the resulting graded poset H has a flag f-vector). 45

50 The dual zonotope Dual to the arrangement H is a zonotope Z = [ z 1, z 1 ] + [ z 2, z 2 ] + + [ z m, z m ] z 2 z 3 z 1 z 1 z 3 z 2 whose lattice of faces F(Z) is Eulerian. Thus F (Z) := F (F(Z)) Π. 46

51 Geometric Lattice Lattice of Regions In the 70 s, Tom Zaslavsky showed how interesting enumerative invariants of arrangements can be obtained from the simpler underlying geometric lattice. For example, the Möbius function of L determines the numbers of regions of H (the f-vector). Later, Bayer-Sturmfels showed that L determines the flag f- vector of H. Recently B, Ehrenborg, Readdy + B, Hsiao, vanwilligenburg made this determination explicit via the descents-to-peaks map ϑ as Theorem: ϑ ( F (L 0) ) = 2 F (Z). 47

52 Labeling Chains in Geometric Lattices L geometric lattice, of rank n, x < y in L cover relation ( no z L, x < z < y) Then y = x a where a is an atom in L (a > 0) Totally order the atoms in L: z 1 < z 2 < < z m or H 1 < H 2 < < H m and label the cover relation x < y by the least i such that the atom a i satisfies y = x a i. In L 0, label the additional cover relation 0 > 0 by a smallest label 0. As a result, every maximal chain C in L 0 has received a sequence l(c) = (a 0, a 1,..., a n ) of labels from the set {0, 1, 2,..., m}, with a 0 = 0. 48

53 Descents and Peaks in l(c) For each chain C in L 0, the label sequence l(c) has a descent set and a peak set. Björner-Stanley (R-labeling): If h S denotes the number of maximal chains C in L 0 with descent set S, then F (L 0) = S h S L S. BER + BHvW: If t S denotes the number of maximal chains C in L 0 with peak set S, then F (Z) = 1 2 S t S Θ S. This is, perhaps, the most explicit description of the relation ϑ ( F (L 0) ) = 2 F (Z). However, a simple understanding of why this works is yet to be. 49

54 Example: Boolean lattice Cube If z i = e i, where e 1, e 2, e 3 are the coordinate vectors in R 3, then L = B 3, the Boolean algebra, and Z = 3-cube (H = coordinate arrangement) The labeling scheme assigns all π S 3 to B 3 and so to B 3 0 the labels 0123, 01 32, 0 213, 02 31, 0 312, So and F (L 0) = L + 2L {2} + 2L {3} + L {2,3}. ϑ ( F (L 0) ) = Θ + 3Θ {2} + 2Θ {3} ( 1 = 2 2 Θ c Θ dc + 4 ) 4 Θ cd cd-index of the 3-cube is c 3 + 6dc + 4cd and that of the coordinate arrangement is (via w w, reversal of cd words) c 3 + 4dc + 6cd. 50

55 Other examples where ϑ (F (Q)) = 2 F (P )? Hsiao: If L is a distributive lattice then ϑ ( F (L 0) ) = 2 F (L) for some shellable Eulerian poset L. This L is the face poset of a regular CW-sphere. More generally (B, Hsiao & Provan), if L is meet-distributive, (closed subsets of an anti-exchange closure [Edelman, et al.]), e.g., the lattice of convex subsets of a finite subset of Euclidean space: ϑ ( F (L 0) ) = 2 F (L) 51

56 Order complex of a distributive lattice Let L be a distributive lattice: L = J(P ), where J(P ) is the lattice of order ideals in a poset P, ordered by inclusion. The order complex (L) is the simplicial complex on the elements of L with simplices being the chains of L. Provan: (L \ ˆ0) can be obtained from a ( P 1)-simplex by a sequence of stellar subdivisions: b <b,c> <c> a P c <b> L=J(P) <a> <a,c> <c> < > <a,c> <a> <b,c> <b> 52

57 To construct the sphere with face poset L: Reflect the order complex into the boundary of the crosspolytope; equivalently, do Provan construction over the boundary of the P -dimensional crosspolytope. Resulting simplicial polytope is the barycentric subdivision (L) of the regular CW-sphere with face poset L. <c> <a,c> < a,c> <a> <b,c> < a> <b> <b, c> <a, c> < a, c> < c> F (L 0) = L + L {2} + L {3} ϑ ( F (L 0) ) = Θ + Θ {2} + Θ {3} = 2F (L) 53

58 Example: convex subsets of three collinear points a < b < c. {a,b,c} {a,b} {b,c} {a} {b} {c} L 0 L F (L 0) = L + L {2} + 2L {3} ϑ ( F (L 0) ) = Θ + Θ {2} + 2Θ {3} = 2F (L ) 54

59 Comments and possible extensions For any oriented matroid with geometric lattice L, ϑ ( F (L 0) ) gives the cd-index of the associated pseudoplane arrangement. Swartz: For nonorientable matroids L, there is an arrangement of homotopy spheres having L as intersection lattice. Conjecture: F (L 0) gives a lower bound for the flag f-vector in this case. (Equality L orientable??) Does ϑ ( F (L 0) ) = 2 F (P ) hold for any semimodular lattice? 55

60 Coxeter Groups (work with F. Brenti) A Coxeter group is a group W generated by a set S with the relations s 2 = e for all s S (e = identity), and otherwise only relations of the form (ss ) m(s,s ) = e, for s s S with m(s, s ) = m(s, s) 2. Examples include the symmetry groups of regular polytopes (and so the symmetric groups) and much more (see Björner-Brenti, Combinatorics of Coxeter Groups, Springer, 2005). 56

61 Bruhat order on (W, S) Each v W can be written v = s 1 s 2 s k with s i S If k is minimal among all such expressions for v, then s 1 s 2 s k is called a reduced expression for v and k = l(v) is called the length of v. Bruhat order on (W, S): if v = s 1 s 2 s k is a reduced expression for v, then u v for u W if some reduced expression for u is a subword u = s i1 s i2 s il of v. Fact: for each u v W the Bruhat interval [u, v] is an Eulerian poset of rank l(v) l(u) 57

62 R-polynomials H(W ) the Hecke algebra associated to W : the free Z[q, q 1 ]- module having the set {T v : v W } as a basis and multiplication such that for all v W and s S: T v T s = { Tvs, if l(vs) > l(v) qt vs + (q 1)T v, if l(vs) < l(v) H(W ) is an associative algebra having T e as unity. invertible in H(W ): for v W, Each T v is (T v 1) 1 = q l(v) u v( 1) l(v) l(u) R u,v (q) T u, where R u,v (q) Z[q]. The polynomials R u,v are called the R-polynomials of W. For u, v W, u v, deg(r u,v ) = l(v) l(u) and R u,u (q) = 1. 58

63 Kazhdan-Lusztig polynomials There is a unique family of polynomials {P u,v (q)} u,v W Z[q], such that, for all u, v W, 1. P u,v (q) = 0 if u v; 2. P u,u (q) = 1; 3. deg(p u,v (q)) 1 2 (l(v) l(u) 1), if u < v; 4. q l(v) l(u) P u,v ( 1 q ) = u z v R u,z (q) P z,v (q), if u v. 59

64 Extended quasisymmetric function of a Bruhat interval For u v W, there exists a (necessarily unique) polynomial R u,v (q) N[q] such that R u,v (q) = q 1 2 (l(v) l(u)) R u,v (q 1 2 q 1 2 ) For Bruhat interval [u, v], the extended quasisymmetric function F ([u, v]) := u=t 0 t 1 t k =v R t0 t 1 (x 1 ) R t1 t 2 (x 2 ) R tk 1 t k (x k ), where, again, the sum is over all multichains in [u, v]. Properties: F ([u 1, v 1 ] [u 2, v 2 ]) = F ([u 1, v 1 ]) F ([u 2, v 2 ]), F ([u, v]) = α c α (u, v) M α = α b α (u, v) L α, where c α and b α count paths in the Bruhat graph (related to Bruhat order [u, v]) 60

65 F ([u, v]) is a peak function Brenti: The c α satisfy the generalized Dehn-Sommerville equations, so we may conclude: F ([u, v]) Π, in fact F ([u, v]) Π l(u,v) Π l(u,v) 2 Π l(u,v) 4 Note: Bruhat order [u, v] is always Eulerian, so F ([u, v]) Π, but usually F ([u, v]) F ([u, v]). In fact F ([u, v]) = F ([u, v]) + lower terms. Since F ([u, v]) Π, we define the extended cd-index of [u, v] by F ([u, v]) = [ ] 1 [w] u,v w 2 w d +1Θ w [w] Θ u,v w = w where the sum is over all cd-words w (with deg(w) = l(u, v) 1, l(u, v) 3,... ). 61

66 Ballot polynomials and Kazhdan-Lusztig polynomials Brenti gave an expresion for P u,v, u < v, in terms of the c α (u, v), ( ) q l(v) l(u) 2 P u,v (q) q l(v) l(u) 1 2 P u,v q = β C b β (u, v) [q β ] 2 Υ β (q) where Υ β (q) enumerates certain implicitly defined lattice paths. By expressing this in terms of the extended cd-index of [u, v], the resulting paths are now explicit, and we can get where P u,v (q) = n/2 i=0 a i q i B n 2i ( q) a i = a i (u, v) = [c n 2i ] u,v + ( 1) w 2 + w d C wd [wdc n 2i ] u,v w B k (q) := k/2 i=0 k+1 2i ( ) k+1 k+1 i q i are the ballot polynomials and C w is a product of Catalan numbers (or 0 if w is not even). 62

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