Variations on the Majorization Order

Transcription

1 Variations on the Majorization Order Curtis Greene, Haverford College Halifax, June 18, 2012

2 A. W. Marshall, I. Olkin, B. C. Arnold Inequalities: Theory of Majorization and its Applications, 2nd Edition Springer 2010 (909 p).

3 G. H. Hardy, J. E. Littlewood, G. Polya Inequalities Cambridge U. Press 1934, 1951, 1967 (324 p).

4 Example: Muirhead s Inequalities (1902) 1 6 (A2 B 2 + A 2 C C 2 D 2 ) 1 12 (A2 BC + A 2 BD + + B 2 CD) Denote this by: Similarly: M 22 M (A5 B 4 C + + B 5 C 4 D) 1 12 (A4 B 4 C B 4 C 2 D 2 ) Denote this by: M 541 M 442. Theorem (Muirhead): M λ M µ λ µ ( Majorization )

5 Majorization: Definition Assume λ and µ are monotone decreasing sequences. Then λ µ iff λ 1 µ 1 λ 1 + λ 2 µ 1 + µ 2 λ 1 + λ 2 + λ 3 µ 1 + µ 2 + λ 3 λ 1 + λ 2 + λ 3 + λ 4 µ 1 + µ 2 + λ 3 + λ 4. etc. Example: λ = {8, 6, 6, 3, 1, 1} {8, 6, 5, 4, 1, 1} = µ Compare partial sums: (λ) (µ)

6 Moving Boxes X X Example: (4, 0, 0, 0) (3, 1, 0, 0) (2, 2, 0, 0) (2, 1, 1, 0) (1, 1, 1, 1) M 4 M 31 M 22 M 211 M 1111

7 The Poset (P 6, ): Partitions of 6

8 The Poset (P 6, ): Partitions of 6 Properties Lattice Self-dual (λ λ ) Möbius function ±1, 0 Non-Properties Not ranked

9 The Poset DP 6 Double-Majorization

10 The Poset DP 6 Double-Majorization What is it? Why interesting? Properties?

11 Why Interesting? Non-homogeneous Muirhead-type inequalities

12 Why Interesting? Non-homogeneous Muirhead-type inequalities Examples: 1 6 (AB+AC+AD+BC+BD+CD) <? > 1 4 (ABC+ABD+ACD+BCD) There can t be an inequality, in either direction.

13 Why Interesting? Non-homogeneous Muirhead-type inequalities Examples: 1 6 (AB+AC+AD+BC+BD+CD) <? > 1 4 (ABC+ABD+ACD+BCD) There can t be an inequality, in either direction. Instead we have ( ) 1 1/2 ( ) 1 1/3 (AB+AC+ +BD+CD) 6 4 (ABC+ABD+ACD+BCD) or ( ) 1 3 ( ) 1 2 (AB+AC+ +BD+CD) 6 4 (ABC+ABD+ACD+BCD)

14 Why Interesting? Non-homogeneous Muirhead-type inequalities Examples: 1 6 (AB+AC+AD+BC+BD+CD) <? > 1 4 (ABC+ABD+ACD+BCD) There can t be an inequality, in either direction. Instead we have ( ) 1 1/2 ( ) 1 1/3 (AB+AC+ +BD+CD) 6 4 (ABC+ABD+ACD+BCD) or ( ) 1 3 ( ) 1 2 (AB+AC+ +BD+CD) 6 4 (ABC+ABD+ACD+BCD) These last inequalities are TRUE (MacLaurin (1729)).

15 Symmetric Monomial Means ( ) 1 1/2 (AB + AC + + BD + CD) = M 11 6 ( ) 1 1/3 (ABC + ABD + ACD + BCD) = M ( ) 1 1/6 6 (A3 B 3 + A 3 C 3 + B 3 C 3 + A 3 D 3 + B 3 D 3 + C 3 D 3 ) = M 33 ( ) 1 1/4 12 (A2 BC + A 2 BD + A 2 CD + + D 2 AC + D 2 BC) = M 211

16 Symmetric Monomial Means ( ) 1 1/2 (AB + AC + + BD + CD) = M 11 6 ( ) 1 1/3 (ABC + ABD + ACD + BCD) = M ( ) 1 1/6 6 (A3 B 3 + A 3 C 3 + B 3 C 3 + A 3 D 3 + B 3 D 3 + C 3 D 3 ) = M 33 ( ) 1 1/4 12 (A2 BC + A 2 BD + A 2 CD + + D 2 AC + D 2 BC) = M 211 Theorems: M 11 M 111 (1729), M 33 M 211 (2009).

17 What is it? Normalized majorization NORMALIZE: λ λ = λ λ Embeds each P n naturally into the lattice (Q 1, ) of nonnegative monotone rational sequences summing to 1, under majorization.

18 What is it? Normalized majorization NORMALIZE: λ λ = λ λ Embeds each P n naturally into the lattice (Q 1, ) of nonnegative monotone rational sequences summing to 1, under majorization. Definition: λ µ if λ µ, i.e., λ λ µ µ. Example: (4, 3, 1, 1, 1) ( 3, 3, 0, 0, 0) (.5,.7,.8,.9, 1) (.5, 1, 1, 1, 1)

19 What is it? Normalized majorization NORMALIZE: λ λ = λ λ Embeds each P n naturally into the lattice (Q 1, ) of nonnegative monotone rational sequences summing to 1, under majorization. Definition: λ µ if λ µ, i.e., λ λ µ µ. Example: (4, 3, 1, 1, 1) ( 3, 3, 0, 0, 0) (.5,.7,.8,.9, 1) (.5, 1, 1, 1, 1) But note that (1, 1, 1) (2, 2, 2) (1, 1, 1). (It s a preorder.)

20 Double Majorization

21 Double Majorization Definition: λ µ iff λ µ and λ µ

22 Double Majorization Definition: Equivalently: λ µ iff λ µ and λ µ λ µ iff λ λ µ µ and λ λ µ µ

23 Double Majorization Definition: Equivalently: λ µ iff λ µ and λ µ λ µ iff λ λ µ µ and λ λ µ µ Example: (Percentages)

24 Double Majorization Definition: Equivalently: λ µ iff λ µ and λ µ λ µ iff λ λ µ µ and λ λ µ µ Example: (Partial sums)

25 Double Majorization Definition: Equivalently: λ µ iff λ µ and λ µ λ µ iff λ λ µ µ and λ λ µ µ Example: (Partial sums) Observe that the conditions λ µ and λ µ are not equivalent.

26 The Poset DP 6 Double-Majorization

27 The Poset DP 6 Double-Majorization What is it? Why interesting? Properties?

28 Application: Muirhead for Symmetric Monomial Means Theorem (Cuttler-Greene-Skandera 2011) If λ and µ are partitions with λ µ, then M λ M µ if and only if λ µ.

29 Application: Muirhead for Symmetric Monomial Means Theorem (Cuttler-Greene-Skandera 2011) If λ and µ are partitions with λ µ, then M λ M µ if and only if λ µ. Conjecture Also true when λ > µ.

30 Application: Muirhead for Symmetric Monomial Means Theorem (Cuttler-Greene-Skandera 2011) If λ and µ are partitions with λ µ, then M λ M µ if and only if λ µ. Conjecture Also true when λ > µ. True for all cases shown in the diagram. Includes several families of classical inequalities (e.g. Maclaurin 1729). Actually, a very strong form of the inequality holds ( y-positivity ). Only if part true for all λ, µ.

31 Properties of DP If λ µ and µ λ, then λ = µ; hence DP is a partial order.

32 Properties of DP If λ µ and µ λ, then λ = µ; hence DP is a partial order. For all n, (P n, ) embeds isomorphically in DP as a subposet.

33 Properties of DP If λ µ and µ λ, then λ = µ; hence DP is a partial order. For all n, (P n, ) embeds isomorphically in DP as a subposet. λ µ if and only if λ µ ; hence DP is self-dual.

34 Properties of DP If λ µ and µ λ, then λ = µ; hence DP is a partial order. For all n, (P n, ) embeds isomorphically in DP as a subposet. λ µ if and only if λ µ ; hence DP is self-dual. DP is not a lattice.

35 Properties of DP If λ µ and µ λ, then λ = µ; hence DP is a partial order. For all n, (P n, ) embeds isomorphically in DP as a subposet. λ µ if and only if λ µ ; hence DP is self-dual. DP is not a lattice. DP is an infinite poset without universal bounds; it is locally finite, but is not locally ranked.

36 Properties of DP If λ µ and µ λ, then λ = µ; hence DP is a partial order. For all n, (P n, ) embeds isomorphically in DP as a subposet. λ µ if and only if λ µ ; hence DP is self-dual. DP is not a lattice. DP is an infinite poset without universal bounds; it is locally finite, but is not locally ranked. All coverings are obtained by adding or moving boxes up, but not always a single box.

37 Coverings in DP If λ = µ, then λ is covered by µ if and only if µ is obtained from λ by moving a single box up.

38 Coverings in DP If λ = µ, then λ is covered by µ if and only if µ is obtained from λ by moving a single box up. If λ is covered by µ with λ < µ, then µ 2 λ.

39 Coverings in DP If λ = µ, then λ is covered by µ if and only if µ is obtained from λ by moving a single box up. If λ is covered by µ with λ < µ, then µ 2 λ. All such coverings are obtained as follows: if λ = a, µ = b with a < b, then: Let S(λ) denote the sequence of partial sums of λ, and let S b (λ) = (b/a)s(λ). Let µ 0 be the unique composition whose partial sum sequence is S b (λ).

40 Coverings in DP If λ = µ, then λ is covered by µ if and only if µ is obtained from λ by moving a single box up. If λ is covered by µ with λ < µ, then µ 2 λ. All such coverings are obtained as follows: if λ = a, µ = b with a < b, then: Let S(λ) denote the sequence of partial sums of λ, and let S b (λ) = (b/a)s(λ). Let µ 0 be the unique composition whose partial sum sequence is S b (λ). Then µ is the unique partition obtained from µ 0 obtained by repeatedly applying (... µ i µ i+1... ) (... µ i+1 µ i... ) if µ i < µ i+1.

41 Coverings in DP If λ = µ, then λ is covered by µ if and only if µ is obtained from λ by moving a single box up. If λ is covered by µ with λ < µ, then µ 2 λ. All such coverings are obtained as follows: if λ = a, µ = b with a < b, then: Let S(λ) denote the sequence of partial sums of λ, and let S b (λ) = (b/a)s(λ). Let µ 0 be the unique composition whose partial sum sequence is S b (λ). Then µ is the unique partition obtained from µ 0 obtained by repeatedly applying (... µ i µ i+1... ) (... µ i+1 µ i... ) if µ i < µ i+1. ( Smort.)

42 Coverings in DP If λ = µ, then λ is covered by µ if and only if µ is obtained from λ by moving a single box up. If λ is covered by µ with λ < µ, then µ 2 λ. All such coverings are obtained as follows: if λ = a, µ = b with a < b, then: Let S(λ) denote the sequence of partial sums of λ, and let S b (λ) = (b/a)s(λ). Let µ 0 be the unique composition whose partial sum sequence is S b (λ). Then µ is the unique partition obtained from µ 0 obtained by repeatedly applying (... µ i µ i+1... ) (... µ i+1 µ i... ) if µ i < µ i+1. ( Smort.) If λ = a, µ = b, a < b and λ covers µ, then µ is (uniquely) obtained by applying the above algorithm to λ.

43 Coverings in DP Example: λ = (1, 1, 1, 1), a = 4, b = 6. S(λ) = (1, 2, 3, 4) (6/4)S(λ) = (3/2, 3, 9/2, 6) S 6 (λ) = (2, 3, 5, 6) µ 0 = (2, 1, 2, 1) µ = (2, 2, 1, 1) Conclusion: λ is covered by µ.

44 Variation: Majorization on Posets Setup: P = finite poset, Z[P] = set maps from P to Z. Identify f Z[P] with the formal sum x P f (x)x. Problem: Given f Z[P], find conditions under which f can be written as a positive linear combination c xy (y x), with c xy 0 x < y. x<y Denote the set of such f s by M(P), and call M(P) the Muirhead cone of P, Solution: f M(P) iff f [K] 0 for all dual order ideals K P and f [P] = 0. Definition (Majorization): f g iff g f M(P).

45 Some Properties (and Non-Properties) Definition: Z n [P] = {f Z[P] f = n} π n [P] = {f Z n [P] f is order-preserving} Both (Z n [P], ) and (π n [P], ) form posets under majorization. The latter are reverse P-partitions.

46 Some Properties (and Non-Properties) Definition: Z n [P] = {f Z[P] f = n} π n [P] = {f Z n [P] f is order-preserving} Theorem: Z n [P] is ranked and self-dual, but in general it is not a lattice. In general, π n (P) is neither ranked nor self-dual, and it is not a lattice. Coverings in both Z n [P] and π n [P] always consist of moving boxes up. In Z[P] it is always a single box, but in π n [P] more than one box may be required α β

47 π n (P) is not a Lattice α β γ δ

48 Another Variation: Principal Majorization If we replace dual order ideals by principal dual order ideals, we get a larger cone M + (P) = {f f [J] 0 for all principal dual order ideals J}, and a new type of majorization: Definition: f p g iff (g f )[J] 0 for all principal dual order ideals J.

49 Another Variation: Principal Majorization If we replace dual order ideals by principal dual order ideals, we get a larger cone M + (P) = {f f [J] 0 for all principal dual order ideals J}, and a new type of majorization: Definition: f p g iff (g f )[J] 0 for all principal dual order ideals J p α β

50 Extreme ray description of the cone M + (P) Theorem: f M + (P) iff f can be expressed as a positive linear combination c z z, with c z 0 z, z P where for all z, z = µ(y, z)y. y z

51 Extreme ray description of the cone M + (P) Theorem: f M + (P) iff f can be expressed as a positive linear combination c z z, with c z 0 z, z P where for all z, z = µ(y, z)y. y z Note: The cone M + (P) contains the cone M(P). It s extreme generators y x are nonnegative linear combinations of the z s. (Standard Möbius function argument.)

52 Papers: References/Acknowledgements Inequalities for Symmetric Means, with Allison Cuttler 07, Mark Skandera (European Jour. Combinatorics, 2011). Inequalities for Symmetric Functions of Degree 3, with Jeffrey Kroll 09, Jonathan Lima 10, Mark Skandera, and Rengyi Xu 12 (in preparation). The Lattice of Two-Rowed Standard Young Tableaux, with Jonathan Lima 10 (in preparation) Mathematica Packages: symfun.m: Julien Colvin 05, Ben Fineman 05, Renggyi (Emily) Xu 12, Ian Burnette 12 posets.m: Eugenie Hunsicker 91, John Dollhopf 94, Sam Hsiao 95, Erica Greene 10, Ian Burnette 12