The Sperner Property

Transkript

1 The Sperner Property Richard P. Stanley M.I.T. and U. Miami January 20, 2019

2 Sperner s theorem Theorem (E. Sperner, 1927). Let S 1,S 2,...,S m be subsets of an n-element set X such that S i S j for i j, Then m ( n n/2 ), achieved by taking all n/2 -element subsets of X.

3 Sperner s theorem Theorem (E. Sperner, 1927). Let S 1,S 2,...,S m be subsets of an n-element set X such that S i S j for i j, Then m ( n n/2 ), achieved by taking all n/2 -element subsets of X. Emanuel Sperner 9 December January 1980

4 Posets A poset (partially ordered set) is a set P with a binary relation satisfying: Reflexivity: t t Antisymmetry: s t, t s s = t Transitivity: s t, t u s u

5 Graded posets chain: u 1 < u 2 < < u k

6 Graded posets chain: u 1 < u 2 < < u k Assume P is finite. P is graded of rank n if P = P 0 P 1 P n, such that every maximal chain has the form t 0 < t 1 < < t n, t i P i.

7 Diagram of a graded poset... P n P P 2 P 1 P n 1 0

8 Rank-symmetry and unimodality Let p i = #P i. Rank-generating function: F P (q) = n p i q i i=0

9 Rank-symmetry and unimodality Let p i = #P i. Rank-generating function: F P (q) = Rank-symmetric: p i = p n i i n p i q i i=0

10 Rank-symmetry and unimodality Let p i = #P i. Rank-generating function: F P (q) = Rank-symmetric: p i = p n i i n p i q i Rank-unimodal: p 0 p 1 p j p j+1 p n for some j i=0

11 Rank-symmetry and unimodality Let p i = #P i. Rank-generating function: F P (q) = Rank-symmetric: p i = p n i i n p i q i Rank-unimodal: p 0 p 1 p j p j+1 p n for some j rank-unimodal and rank-symmetric j = n/2 i=0

12 The Sperner property antichain A P: s,t A, s t s = t

13 The Sperner property antichain A P: s,t A, s t s = t Note. P i is an antichain

14 The Sperner property antichain A P: Note. P i is an antichain s,t A, s t s = t P is Sperner (or has the Sperner property) if max A #A = maxp i i

15 An example rank-symmetric, rank-unimodal, F P (q) = 3+3q

16 An example rank-symmetric, rank-unimodal, F P (q) = 3+3q not Sperner

17 The boolean algebra B n : subsets of {1,2,...,n}, ordered by inclusion

18 The boolean algebra B n : subsets of {1,2,...,n}, ordered by inclusion p i = ( n) i, FBn (q) = (1+q) n rank-symmetric, rank-unimodal

19 Diagram of B φ

20 Sperner s theorem, restated Theorem. The boolean algebra B n is Sperner. Proof (D. Lubell, 1966). B n has n! maximal chains.

21 Sperner s theorem, restated Theorem. The boolean algebra B n is Sperner. Proof (D. Lubell, 1966). B n has n! maximal chains. If S B n and #S = i, then i!(n i)! maximal chains of B n contain S.

22 Sperner s theorem, restated Theorem. The boolean algebra B n is Sperner. Proof (D. Lubell, 1966). B n has n! maximal chains. If S B n and #S = i, then i!(n i)! maximal chains of B n contain S. Let A be an antichain. Since a maximal chain can intersect at most one element of A, we have S!(n S )! n!. S A

23 Sperner s theorem, restated Theorem. The boolean algebra B n is Sperner. Proof (D. Lubell, 1966). B n has n! maximal chains. If S B n and #S = i, then i!(n i)! maximal chains of B n contain S. Let A be an antichain. Since a maximal chain can intersect at most one element of A, we have S!(n S )! n!. S A Divide by n!: S A 1 ( n ) 1. S

24 Lubell s proof (cont.) Divide by n!: S A 1 ( n ) 1. S

25 Lubell s proof (cont.) Divide by n!: Now ( n ) ( S n n/2 ), so S A A S 1 ( n ) 1. S 1 ( n ) 1. n/2

26 Lubell s proof (cont.) Divide by n!: Now ( n ) ( S n n/2 ), so A ( ) n n/2 S A A S 1 ( n ) 1. S 1 ( n ) 1. n/2

27 A q-analogue Lubell s proof carries over to some other posets. Theorem. Let B n (q) denote the poset of all subspaces of F n q. Then B n (q) is Sperner.

28 Linear algebra P = P 0 P m : graded poset QP i : vector space with basis Q U: QP i QP i+1 is order-raising if for all s P i, U(s) span Q {t P i+1 : s < t}

29 Order-matchings Order matching: µ: P i P i+1 : injective and µ(t) < t

30 Order-matchings Order matching: µ: P i P i+1 : injective and µ(t) < t P i +1 P i µ

31 Order-raising and order-matchings Key Lemma. If U: QP i QP i+1 is injective and order-raising, then there exists an order-matching µ: P i P i+1.

32 Order-raising and order-matchings Key Lemma. If U: QP i QP i+1 is injective and order-raising, then there exists an order-matching µ: P i P i+1. Proof. Consider the matrix of U with respect to the bases P i and P i+1.

33 Key lemma proof P i s 1. s m P i+1 {}}{ t 1 t m t n det 0

34 Key lemma proof P i s 1. s m P i+1 {}}{ t 1 t m t n det 0 s 1 < t 1,...,s m < t m

35 Minor variant Similarly if there exists surjective order-raising U: QP i QP i+1, then there exists an order-matching µ: P i+1 P i.

36 A criterion for Spernicity P = P 0 P n : finite graded poset Proposition. If for some j there exist order-raising operators QP 0 inj. QP 1 inj. inj. QP j surj. QP j+1 surj. surj. QP n, then P is rank-unimodal and Sperner.

37 A criterion for Spernicity P = P 0 P n : finite graded poset Proposition. If for some j there exist order-raising operators QP 0 inj. QP 1 inj. inj. QP j surj. QP j+1 surj. surj. QP n, then P is rank-unimodal and Sperner. Proof. Rank-unimodal clear: p 0 p 1 p j p j+1 p n.

38 A criterion for Spernicity P = P 0 P n : finite graded poset Proposition. If for some j there exist order-raising operators QP 0 inj. QP 1 inj. inj. QP j surj. QP j+1 surj. surj. QP n, then P is rank-unimodal and Sperner. Proof. Rank-unimodal clear: p 0 p 1 p j p j+1 p n. Glue together the order-matchings.

39 Gluing example j

40 Gluing example j

41 Gluing example j

42 Gluing example j

43 Gluing example j

44 Gluing example j

45 A chain decomposition P = C 1 C pj (chains) A = antichain,c = chain #(A C) 1 #A p j.

46 Back to B n Explicit order matching (B n ) i (B n ) i+1 for i < n/2: Example. S = {1,4,6,7,11} (B 13 ) 5

47 Back to B n Explicit order matching (B n ) i (B n ) i+1 for i < n/2: Example. S = {1,4,6,7,11} (B 13 )

48 Back to B n Explicit order matching (B n ) i (B n ) i+1 for i < n/2: Example. S = {1,4,6,7,11} (B 13 ) 5 ) ) ) ) )

49 Back to B n Explicit order matching (B n ) i (B n ) i+1 for i < n/2: Example. S = {1,4,6,7,11} (B 13 ) 5 ) ( ( ) ( ) ) ( ( ( ) ( (

50 Back to B n Explicit order matching (B n ) i (B n ) i+1 for i < n/2: Example. S = {1,4,6,7,11} (B 13 ) 5 ) ( ( ) ( ) ) ( ( ( ) ( (

51 Back to B n Explicit order matching (B n ) i (B n ) i+1 for i < n/2: Example. S = {1,4,6,7,11} (B 13 ) 5 ) ( ( ) ( ) ) ( ( ( ) ( (

52 Back to B n Explicit order matching (B n ) i (B n ) i+1 for i < n/2: Example. S = {1,4,6,7,11} (B 13 ) 5 ) ( ( ) ( ) ) ( ( ( ) ( (

53 Order-raising for B n Define by U: Q(B n ) i Q(B n ) i+1 U(S) = #T =i+1 S T T, S (B n ) i.

54 Order-raising for B n Define by U: Q(B n ) i Q(B n ) i+1 U(S) = #T =i+1 S T T, S (B n ) i. Similarly define D: Q(B n ) i+1 Q(B n ) i by D(T) = #S=i S T S, T (B n ) i+1.

55 Order-raising for B n Define by U: Q(B n ) i Q(B n ) i+1 U(S) = #T =i+1 S T T, S (B n ) i. Similarly define D: Q(B n ) i+1 Q(B n ) i by D(T) = #S=i S T S, T (B n ) i+1. Note. UD is positive semidefinite, and hence has nonnegative real eigenvalues, since the matrices of U and D with respect to the bases (B n ) i and (B n ) i+1 are transposes.

56 A commutation relation Lemma. On Q(B n ) i we have where I is the identity operator. DU UD = (n 2i)I,

57 A commutation relation Lemma. On Q(B n ) i we have where I is the identity operator. DU UD = (n 2i)I, Corollary. If i < n/2 then U is injective. Proof. UD has eigenvalues θ 0, and eigenvalues of DU are θ +n 2i > 0. Hence DU is invertible, so U is injective. Similarly U is surjective for i n/2.

58 A commutation relation Lemma. On Q(B n ) i we have where I is the identity operator. DU UD = (n 2i)I, Corollary. If i < n/2 then U is injective. Proof. UD has eigenvalues θ 0, and eigenvalues of DU are θ +n 2i > 0. Hence DU is invertible, so U is injective. Similarly U is surjective for i n/2. Corollary. B n is Sperner.

59 What s the point?

60 What s the point? The symmetric group S n acts on B n by w {a 1,...,a k } = {w a 1,...,w a k }. If G is a subgroup of S n, define the quotient poset B n /G to be the poset on the orbits of G (acting on B n ), with o o S o,t o, S T.

61 An example n = 3, G = {(1)(2)(3),(1,2)(3)} , ,2 φ

62 Spernicity of B n /G Easy: B n /G is graded of rank n and rank-symmetric.

63 Spernicity of B n /G Easy: B n /G is graded of rank n and rank-symmetric. Theorem. B n /G is rank-unimodal and Sperner.

64 Spernicity of B n /G Easy: B n /G is graded of rank n and rank-symmetric. Theorem. B n /G is rank-unimodal and Sperner. Crux of proof. The action of w G on B n commutes with U, so we can transfer U to B n /G, preserving injectivity on the bottom half.

65 An interesting example R: set of squares of an m n rectangle of squares. G mn S R : can permute elements in each row, and permute rows among themselves, so #G mn = n! m m!. G mn = Sn S m (wreath product)

66 Young diagrams A subset S R is a Young diagram if it is left-justified, with weakly decreasing row lengths from top to bottom.

67 Young diagrams A subset S R is a Young diagram if it is left-justified, with weakly decreasing row lengths from top to bottom.

68 B R /G mn Lemma. Each orbit o B R /G mn contains exactly one Young diagram.

69 B R /G mn Lemma. Each orbit o B R /G mn contains exactly one Young diagram. Proof. Let S B R. Let α i be the number of elements of S in row i. Let λ 1 λ m be the decreasing rearrangement of α 1,...,α m. Then the unique Young diagram in the orbit containing S has λ i elements in row i.

70 Poset structure of B r /G mn Y o : Young diagram in the orbit o Easy : o o in B R /G mn if and only if Y o Y o (containment of Young diagram).

71 Poset structure of B r /G mn Y o : Young diagram in the orbit o Easy : o o in B R /G mn if and only if Y o Y o (containment of Young diagram). L(m, n): poset of Young diagrams in an m n rectangle Corollary. B R /G mn = L(m,n)

72 Examples of L(m, n) φ φ 1 φ

73 L(3, 3) φ

74 q-binomial coefficients For 0 k n, define the q-binomial coefficient [ n k] = (1 qn )(1 q n 1 ) (1 q n k+1 ) (1 q k )(1 q k 1. ) (1 q)

75 q-binomial coefficients For 0 k n, define the q-binomial coefficient Example. [ n = k] (1 qn )(1 q n 1 ) (1 q n k+1 ) (1 q k )(1 q k 1. ) (1 q) [ 4 2] = 1+q +2q 2 +q 3 +q 4

76 Properties [ n k ] N[q]

77 Properties [ n ] k N[q] [ n k]q=1 = ( n) k

78 Properties [ n ] k N[q] [ n k]q=1 = ( n) k If q is a prime power, [ n k] is the number of k-dimensional subspaces of F n q (irrelevant here).

79 Properties [ n ] k N[q] [ n k]q=1 = ( n) k If q is a prime power, [ n k] is the number of k-dimensional subspaces of F n q (irrelevant here). F(L(m,n),q) = [ m+n m ]

80 Unimodality Corollary. [ m+n] m has unimodal coefficients.

81 Unimodality Corollary. [ m+n] m has unimodal coefficients. First proved by J. J. Sylvester (1878) using invariant theory of binary forms.

82 Unimodality Corollary. [ m+n] m has unimodal coefficients. First proved by J. J. Sylvester (1878) using invariant theory of binary forms. Combinatorial proof by K. O Hara (1990): explicit injection L(m,n) i L(m,n) i+1, 0 i < 1 2 mn.

83 Unimodality Corollary. [ m+n] m has unimodal coefficients. First proved by J. J. Sylvester (1878) using invariant theory of binary forms. Combinatorial proof by K. O Hara (1990): explicit injection L(m,n) i L(m,n) i+1, 0 i < 1 2 mn. Not an order-matching. Still open to find an explicit order-matching L(m,n) i L(m,n) i+1.

84 Algebraic geometry X: smooth complex projective variety of dimension n H (X;C) = H 0 (X;C) H 1 (X;C) H 2n (X;C): cohomology ring, so H i = H 2n i. Hard Lefschetz Theorem. There exists ω H 2 (the class of a generic hyperplane section) such that for 0 i n, the map ω n 2i : H i H 2n i is a bijection. Thus ω: H i H i+1 is injective for i n and surjective for i n.

85 Cellular decompositions X has a cellular decomposition if X = C i, each C i = C d i (as affine varieties), and each C i is a union of C j s.

86 Cellular decompositions X has a cellular decomposition if X = C i, each C i = C d i (as affine varieties), and each C i is a union of C j s. Fact. If X has a cellular decomposition and [C i ] H 2(n i) denotes the corresponding cohomology classes, then the [C i ] s form a C-basis for H.

87 The cellular decomposition poset Let X = C i be a cellular decomposition. Define a poset P X = {C i }, by C i C j if C i C j (closure in Zariski or classical topology).

88 The cellular decomposition poset Let X = C i be a cellular decomposition. Define a poset P X = {C i }, by C i C j if C i C j (closure in Zariski or classical topology). Easy: P X is graded of rank n. #(P X ) i = dim C H 2(n i) (X;C)

89 The cellular decomposition poset Let X = C i be a cellular decomposition. Define a poset P X = {C i }, by C i C j if C i C j (closure in Zariski or classical topology). Easy: P X is graded of rank n. #(P X ) i = dim C H 2(n i) (X;C) P X is rank-symmetric (Poincaré duality)

90 The cellular decomposition poset Let X = C i be a cellular decomposition. Define a poset P X = {C i }, by C i C j if C i C j (closure in Zariski or classical topology). Easy: P X is graded of rank n. #(P X ) i = dim C H 2(n i) (X;C) P X is rank-symmetric (Poincaré duality) P X is rank-unimodal (hard Lefschetz)

91 Spernicity of P X Identify CP with H (X;C) via C i [C i ].

92 Spernicity of P X Identify CP with H (X;C) via C i [C i ]. Recall: ω H 2 (X;C) (class of hyperplane section) Interpretation of cup product on H (X;C) as intersection implies that ω is order-raising. Hard Lefschetz ω: H 2i H 2(i+1) is injective for i < n/2 and surjective for i n/2.

93 Spernicity of P X Identify CP with H (X;C) via C i [C i ]. Recall: ω H 2 (X;C) (class of hyperplane section) Interpretation of cup product on H (X;C) as intersection implies that ω is order-raising. Hard Lefschetz ω: H 2i H 2(i+1) is injective for i < n/2 and surjective for i n/2. Theorem. P X has the Sperner property.

94 Main example What smooth projective varieties have cellular decompositions?

95 Main example What smooth projective varieties have cellular decompositions? Generalized flag variety: G/Q, where G is a semisimple algebraic group over C, and Q is a parabolic subgroup

96 Main example What smooth projective varieties have cellular decompositions? Generalized flag variety: G/Q, where G is a semisimple algebraic group over C, and Q is a parabolic subgroup Example. Gr(n,k) = SL(n,C)/Q for a certain Q, the Grassmann variety of all k-dimensional subspaces of C n.

97 Main example What smooth projective varieties have cellular decompositions? Generalized flag variety: G/Q, where G is a semisimple algebraic group over C, and Q is a parabolic subgroup Example. Gr(n,k) = SL(n,C)/Q for a certain Q, the Grassmann variety of all k-dimensional subspaces of C n. rational canonical form P Gr(m+n,m) = L(m,n)

98 Best special case G = SO(2n +1,C), Q = spin maximal parabolic subgroup M(n) := P G/Q = Bn /S n, where B n is the hyperoctahedral group (symmetries of n-cube) of order 2 n n!, so #M(n) = 2 n

99 Best special case G = SO(2n +1,C), Q = spin maximal parabolic subgroup M(n) := P G/Q = Bn /S n, where B n is the hyperoctahedral group (symmetries of n-cube) of order 2 n n!, so #M(n) = 2 n M(n) is isomorphic to the set of all subsets of {1,2,...,n} with the ordering {a 1 > a 2 > > a r } {b 1 > b 2 > > b s }, if r s and a i b i for 1 i r.

100 Examples of M(n) φ 1 M(1) φ M(2) φ 2 M(3) 1 φ M(4)

101 Rank-generating function of M(n) rank of {a 1,...,a r } in M(n) is a i

102 Rank-generating function of M(n) rank of {a 1,...,a r } in M(n) is a i ( n 2) F(M(n),q) := M(n) i q i = (1+q)(1+q 2 ) (1+q n ) i=0 Corollary. The polynomial (1+q)(1+q 2 ) (1+q n ) has unimodal coefficients.

103 Rank-generating function of M(n) rank of {a 1,...,a r } in M(n) is a i ( n 2) F(M(n),q) := M(n) i q i = (1+q)(1+q 2 ) (1+q n ) i=0 Corollary. The polynomial (1+q)(1+q 2 ) (1+q n ) has unimodal coefficients. No combinatorial proof known, though can be done with just elementary linear algebra (Proctor).

104 The function f(s,α) Let S R, #S <, α R. f(s,α) = #{T S : i T i = α} Note. i i = 0

105 The function f(s,α) Let S R, #S <, α R. f(s,α) = #{T S : i T i = α} Note. i i = 0 Example. f({1,2,4,5,7,10},7) = 3: 7 = 2+5 = 1+2+4

106 The Erdős-Moser conjecture for R + Let R + = {i R : i > 0}. Erdős-Moser Conjecture for R + S R +, #S = n ( ( ) ) 1 n+1 f(s,α) f {1,2,...,n}, 2 2

107 The Erdős-Moser conjecture for R + Let R + = {i R : i > 0}. Erdős-Moser Conjecture for R + S R +, #S = n ( ( ) ) 1 n+1 f(s,α) f {1,2,...,n}, 2 2 Note. 1 ( n+1 ) 2 2 = 1 2 (1+2+ +n)

108 The proof Proof. Suppose S = {a 1,...,a k }, a 1 > > a k. Let a i1 + +a ir = a j1 + +a js, where i 1 > > i r, j 1 > > j s.

109 The proof Proof. Suppose S = {a 1,...,a k }, a 1 > > a k. Let a i1 + +a ir = a j1 + +a js, where i 1 > > i r, j 1 > > j s. Now {i 1,...,i r } {j 1,...,j s } in M(n) r s, i 1 j 1,...,i s j s a i1 b j1,...,a is b js r = s, a ik = b ik k.

110 Conclusion of proof Thus a i1 + +a ir = b j1 + +b js {i 1,...,i r } and {j 1,...,j s } are incomparable or equal in M(n) ( ( ) ) 1 n+1 #S max #A = f {1,...,n}, A 2 2

111 The weak order on S n s i :== (i,i +1) S n, 1 i n 1 (adjacent transposition) For w S n, l(w) := #{(i,j) : i < j,w(i) > w(j)} = min{p : w = s i1 s ip }.

112 The weak order on S n s i :== (i,i +1) S n, 1 i n 1 (adjacent transposition) For w S n, l(w) := #{(i,j) : i < j,w(i) > w(j)} = min{p : w = s i1 s ip }. weak (Bruhat) order W n on S n : u < v if v = us i1 s ip, p = l(v) l(u).

113 The weak order on S n s i :== (i,i +1) S n, 1 i n 1 (adjacent transposition) For w S n, l(w) := #{(i,j) : i < j,w(i) > w(j)} = min{p : w = s i1 s ip }. weak (Bruhat) order W n on S n : u < v if v = us i1 s ip, p = l(v) l(u). l is the rank function of W n, so F(W n,q) = (1+q)(1+q +q 2 ) (1+q + +q n 1 ).

114 The weak order on S n s i :== (i,i +1) S n, 1 i n 1 (adjacent transposition) For w S n, l(w) := #{(i,j) : i < j,w(i) > w(j)} = min{p : w = s i1 s ip }. weak (Bruhat) order W n on S n : u < v if v = us i1 s ip, p = l(v) l(u). l is the rank function of W n, so F(W n,q) = (1+q)(1+q +q 2 ) (1+q + +q n 1 ). A. Björner (1984): does W n have the Sperner property?

115 Examples of weak order W W 4

116 An order-raising operator theory of Schubert polynomials suggests: U(w) := i ws i 1 i n 1 ws i >s i

117 An order-raising operator theory of Schubert polynomials suggests: U(w) := i ws i 1 i n 1 ws i >s i Fact (Macdonald, Fomin-S). Let u < v in W n, l(v) l(u) = p. The coefficient of v in U p (u) is p!s vu 1(1,1,...,1), where S w (x 1,...,x n 1 ) is a Schubert polynomial.

118 A down operator C. Gaetz and Y. Gao (2018): constructed D: Q(W n ) i Q(W n ) i 1 such that (( ) ) n DU UD = 2i I. 2 Suffices for Spernicity.

119 A down operator C. Gaetz and Y. Gao (2018): constructed D: Q(W n ) i Q(W n ) i 1 such that (( ) ) n DU UD = 2i I. 2 Suffices for Spernicity. Note. D is order-lowering on the strong Bruhat order. Leads to duality between weak and strong order.

120 Another method Z. Hamaker, O. Pechenik, D. Speyer, and A. Weigandt (2018): for k < 1 2( n 2), let D(n, k) = matrix of U (n 2) 2k : Q(W n ) k Q(W n ) ( n 2) k with respect to the bases (W n ) k and (W n ) ( n Then (conjectured by RS): detd(n,k) = ± (( ) n 2k )! #(Wn) k 1 k 2 i=0 2) k (( n 2 (in some order). ) ) #(Wn) i (k +i). k i

121 Another method Z. Hamaker, O. Pechenik, D. Speyer, and A. Weigandt (2018): for k < 1 2( n 2), let D(n, k) = matrix of U (n 2) 2k : Q(W n ) k Q(W n ) ( n 2) k with respect to the bases (W n ) k and (W n ) ( n Then (conjectured by RS): detd(n,k) = ± (( ) n 2k )! #(Wn) k 1 k 2 i=0 2) k (( n 2 (in some order). ) ) #(Wn) i (k +i). k i Also suffices to prove Sperner property (just need detd(n,k) 0).

122 An open problem The weak order W(G) can be defined for any (finite) Coxeter group G. Is W(G) Sperner?

123 The final slide

124 The final slide