Comparing skew Schur functions: a quasisymmetric perspective Peter - - PowerPoint PPT Presentation

Comparing skew Schur functions: a quasisymmetric perspective

Peter McNamara Bucknell University AMS/EMS/SPM International Meeting 11 June 2015 Slides and paper available from www.facstaff.bucknell.edu/pm040/

Comparing skew Schur functions quasisymmetrically Peter McNamara 1

Outline

◮ The background story: the equality question ◮ Conditions for Schur-positivity ◮ Quasisymmetric insights and the main conjecture

Comparing skew Schur functions quasisymmetrically Peter McNamara 2

Preview

F-support containment Dual of row overlap dominance

Comparing skew Schur functions quasisymmetrically Peter McNamara 3

Schur functions

Cauchy, 1815

◮ Partition λ = (λ1, λ2, . . . , λℓ) ◮ Young diagram.

Example: λ = (4, 4, 3, 1)

7 4 1 3 3 4 9 4 4 6 6 5

Comparing skew Schur functions quasisymmetrically Peter McNamara 4

Schur functions

Cauchy, 1815

◮ Partition λ = (λ1, λ2, . . . , λℓ) ◮ Young diagram.

Example: λ = (4, 4, 3, 1)

◮ Semistandard Young tableau

(SSYT)

6 3 3 4 9 1 5 7 4 4 4 6

Comparing skew Schur functions quasisymmetrically Peter McNamara 4

Schur functions

Cauchy, 1815

◮ Partition λ = (λ1, λ2, . . . , λℓ) ◮ Young diagram.

Example: λ = (4, 4, 3, 1)

◮ Semistandard Young tableau

(SSYT)

6 3 3 4 9 1 5 7 4 4 4 6

The Schur function sλ in the variables x = (x1, x2, . . .) is then defined by sλ =

• SSYT T

x#1’s in T

1

x#2’s in T

2

· · · . Example. s4431 = x1x2

3x4 4x5x2 6x7x9 + · · · .

Comparing skew Schur functions quasisymmetrically Peter McNamara 4

Skew Schur functions

Cauchy, 1815

◮ Partition λ = (λ1, λ2, . . . , λℓ) ◮ µ fits inside λ. ◮ Young diagram.

Example: λ/µ = (4, 4, 3, 1)/(3, 1)

◮ Semistandard Young tableau

(SSYT)

4 9 5 7 4 4 6 6

The skew Schur function sλ/µ in the variables x = (x1, x2, . . .) is then defined by sλ/µ =

• SSYT T

x#1’s in T

1

x#2’s in T

2

· · · . Example. s4431/31 = x3

4x5x2 6x7x9 + · · · .

Comparing skew Schur functions quasisymmetrically Peter McNamara 4

The beginning of the story

sA: the skew Schur function for the skew shape A. Key Facts.

◮ sA is symmetric in the variables x1, x2, . . .. ◮ The (non-skew) sλ form a basis for the symmetric functions.

Comparing skew Schur functions quasisymmetrically Peter McNamara 5

The beginning of the story

sA: the skew Schur function for the skew shape A. Key Facts.

◮ sA is symmetric in the variables x1, x2, . . .. ◮ The (non-skew) sλ form a basis for the symmetric functions.

Wide Open Question. When is sA = sB? Determine necessary and sufficient conditions on shapes of A and B. = =

Comparing skew Schur functions quasisymmetrically Peter McNamara 5

The beginning of the story

sA: the skew Schur function for the skew shape A. Key Facts.

◮ sA is symmetric in the variables x1, x2, . . .. ◮ The (non-skew) sλ form a basis for the symmetric functions.

Wide Open Question. When is sA = sB? Determine necessary and sufficient conditions on shapes of A and B. = =

◮ Lou Billera, Hugh Thomas, Steph van Willigenburg (2004) ◮ John Stembridge (2004) ◮ Vic Reiner, Kristin Shaw, Steph van Willigenburg (2006) ◮ McN., Steph van Willigenburg (2006) ◮ Christian Gutschwager (2008)

Comparing skew Schur functions quasisymmetrically Peter McNamara 5

Necessary conditions for equality

Comparing skew Schur functions quasisymmetrically Peter McNamara 6

Necessary conditions for equality

General idea: the overlaps among rows must match up.

Comparing skew Schur functions quasisymmetrically Peter McNamara 6

Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A, let

• verlapk(i) be the number of columns occupied in common by rows

i, i + 1, . . . , i + k − 1. Then rowsk(A) is the weakly decreasing rearrangement of (overlapk(1), overlapk(2), . . .). Example. A =

Comparing skew Schur functions quasisymmetrically Peter McNamara 6

Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A, let

• verlapk(i) be the number of columns occupied in common by rows

i, i + 1, . . . , i + k − 1. Then rowsk(A) is the weakly decreasing rearrangement of (overlapk(1), overlapk(2), . . .). Example. A =

◮ overlap1(i) = length of the ith row. Thus rows1(A) = 44211.

Comparing skew Schur functions quasisymmetrically Peter McNamara 6

Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A, let

• verlapk(i) be the number of columns occupied in common by rows

i, i + 1, . . . , i + k − 1. Then rowsk(A) is the weakly decreasing rearrangement of (overlapk(1), overlapk(2), . . .). Example. A =

◮ overlap1(i) = length of the ith row. Thus rows1(A) = 44211. ◮ overlap2(1) = 2, overlap2(2) = 3, overlap2(3) = 1,

• verlap2(4) = 1,

so rows2(A) = 3211.

Comparing skew Schur functions quasisymmetrically Peter McNamara 6

Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A, let

• verlapk(i) be the number of columns occupied in common by rows

i, i + 1, . . . , i + k − 1. Then rowsk(A) is the weakly decreasing rearrangement of (overlapk(1), overlapk(2), . . .). Example. A =

◮ overlap1(i) = length of the ith row. Thus rows1(A) = 44211. ◮ overlap2(1) = 2, overlap2(2) = 3, overlap2(3) = 1,

• verlap2(4) = 1,

so rows2(A) = 3211.

◮ rows3(A) = 11.

Comparing skew Schur functions quasisymmetrically Peter McNamara 6

Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A, let

• verlapk(i) be the number of columns occupied in common by rows

i, i + 1, . . . , i + k − 1. Then rowsk(A) is the weakly decreasing rearrangement of (overlapk(1), overlapk(2), . . .). Example. A =

◮ overlap1(i) = length of the ith row. Thus rows1(A) = 44211. ◮ overlap2(1) = 2, overlap2(2) = 3, overlap2(3) = 1,

• verlap2(4) = 1,

so rows2(A) = 3211.

◮ rows3(A) = 11. ◮ rowsk(A) = ∅ for k > 3.

Comparing skew Schur functions quasisymmetrically Peter McNamara 6

Necessary conditions for equality

Theorem [RSvW, 2006]. Let A and B be skew shapes. If sA = sB, then rowsk(A) = rowsk(B) for all k.

Comparing skew Schur functions quasisymmetrically Peter McNamara 7

Necessary conditions for equality

Theorem [RSvW, 2006]. Let A and B be skew shapes. If sA = sB, then rowsk(A) = rowsk(B) for all k. supps(A): Schur support of A supps(A) = {λ : sλ appears in Schur expansion of sA}

• Example. A =

sA = s3 + 2s21 + s111 supps(A) = {3, 21, 111}.

Comparing skew Schur functions quasisymmetrically Peter McNamara 7

Necessary conditions for equality

Theorem [RSvW, 2006]. Let A and B be skew shapes. If sA = sB, then rowsk(A) = rowsk(B) for all k. supps(A): Schur support of A supps(A) = {λ : sλ appears in Schur expansion of sA}

• Example. A =

sA = s3 + 2s21 + s111 supps(A) = {3, 21, 111}. Theorem [McN., 2008]. It suffices to assume supps(A) = supps(B).

Comparing skew Schur functions quasisymmetrically Peter McNamara 7

Necessary conditions for equality

Theorem [RSvW, 2006]. Let A and B be skew shapes. If sA = sB, then rowsk(A) = rowsk(B) for all k. supps(A): Schur support of A supps(A) = {λ : sλ appears in Schur expansion of sA}

• Example. A =

sA = s3 + 2s21 + s111 supps(A) = {3, 21, 111}. Theorem [McN., 2008]. It suffices to assume supps(A) = supps(B). Converse is definitely not true: =

Comparing skew Schur functions quasisymmetrically Peter McNamara 7

Schur-positivity order

Our interest: inequalities. Skew Schur functions are Schur-positive: sλ/µ =

• ν

cλ

µνsν.

Original Question. When is sλ/µ − sσ/τ Schur-positive?

Comparing skew Schur functions quasisymmetrically Peter McNamara 8

Schur-positivity order

Our interest: inequalities. Skew Schur functions are Schur-positive: sλ/µ =

• ν

cλ

µνsν.

Original Question. When is sλ/µ − sσ/τ Schur-positive?

• Definition. Let A, B be skew shapes. We say that

A ≥s B if sA − sB is Schur-positive. Original goal: Characterize the Schur-positivity order ≥s in terms of skew shapes.

Comparing skew Schur functions quasisymmetrically Peter McNamara 8

Example of a Schur-positivity poset

If B ≤s A then |A| = |B|. Call the resulting

• rdered set Pn.

Then P4:

Comparing skew Schur functions quasisymmetrically Peter McNamara 9

More examples

P5: P6:

XXXXX XXXX X XXXX X XXX XX XXX XX XXX XX XXX X X XXX X X XXX X X X XXX X XXX X X XX XX X XX XX X XX XX X XX XX X XX XX X XX XX X XX X XX XX X X X XX X X X XX X X X X XX X X XX X X X XX X X X XX X X X X XX X X XX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

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Known properties: Sufficient conditions

Sufficient conditions for A ≥s B:

◮ Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon (1997) ◮ Andrei Okounkov (1997) ◮ Sergey Fomin, William Fulton, Chi-Kwong Li, Yiu-Tung Poon

(2003)

◮ Anatol N. Kirillov (2004) ◮ Thomas Lam, Alex Postnikov, Pavlo Pylyavskyy (2005) ◮ François Bergeron, Riccardo Biagioli, Mercedes Rosas (2006) ◮ McN., Steph van Willigenburg (2009, 2012) ◮ ...

Comparing skew Schur functions quasisymmetrically Peter McNamara 11

Necessary conditions for Schur-positivity

Comparing skew Schur functions quasisymmetrically Peter McNamara 12

Necessary conditions for Schur-positivity

• Notation. Write λ µ if λ is less than or equal to µ in dominance
• rder, i.e.

λ1 + · · · λi ≤ µ1 + · · · µi for all i.

Comparing skew Schur functions quasisymmetrically Peter McNamara 12

Necessary conditions for Schur-positivity

• Notation. Write λ µ if λ is less than or equal to µ in dominance
• rder, i.e.

λ1 + · · · λi ≤ µ1 + · · · µi for all i. Theorem [McN. (2008)]. Let A and B be skew shapes. If sA − sB is Schur-positive, then rowsk(A) rowsk(B) for all k.

Comparing skew Schur functions quasisymmetrically Peter McNamara 12

Necessary conditions for Schur-positivity

• Notation. Write λ µ if λ is less than or equal to µ in dominance
• rder, i.e.

λ1 + · · · λi ≤ µ1 + · · · µi for all i. Theorem [McN. (2008)]. Let A and B be skew shapes. If sA − sB is Schur-positive, then rowsk(A) rowsk(B) for all k. In fact, it suffices to assume that supps(A) ⊇ supps(B).

Comparing skew Schur functions quasisymmetrically Peter McNamara 12

Necessary conditions for Schur-positivity

• Notation. Write λ µ if λ is less than or equal to µ in dominance
• rder, i.e.

λ1 + · · · λi ≤ µ1 + · · · µi for all i. Theorem [McN. (2008)]. Let A and B be skew shapes. If sA − sB is Schur-positive, then rowsk(A) rowsk(B) for all k. In fact, it suffices to assume that supps(A) ⊇ supps(B). Application. A = B = rows1(A) = 3221 rows1(B) = 2221

Comparing skew Schur functions quasisymmetrically Peter McNamara 12

Necessary conditions for Schur-positivity

• Notation. Write λ µ if λ is less than or equal to µ in dominance
• rder, i.e.

λ1 + · · · λi ≤ µ1 + · · · µi for all i. Theorem [McN. (2008)]. Let A and B be skew shapes. If sA − sB is Schur-positive, then rowsk(A) rowsk(B) for all k. In fact, it suffices to assume that supps(A) ⊇ supps(B). Application. A = B = rows1(A) = 3221 rows1(B) = 2221 rows2(B) = 21 rows2(A) = 111

Comparing skew Schur functions quasisymmetrically Peter McNamara 12

Necessary conditions for Schur-positivity

• Notation. Write λ µ if λ is less than or equal to µ in dominance
• rder, i.e.

λ1 + · · · λi ≤ µ1 + · · · µi for all i. Theorem [McN. (2008)]. Let A and B be skew shapes. If sA − sB is Schur-positive, then rowsk(A) rowsk(B) for all k. In fact, it suffices to assume that supps(A) ⊇ supps(B). Application. A = B = rows1(A) = 3221 rows1(B) = 2221 rows2(B) = 21 rows2(A) = 111 So A and B are incomparable in Schur-positivity poset (and in “Schur support containment poset”).

Comparing skew Schur functions quasisymmetrically Peter McNamara 12

Summary so far

sA − sB is Schur-pos. supps(A) ⊇ supps(B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒

Comparing skew Schur functions quasisymmetrically Peter McNamara 13

Summary so far

sA − sB is Schur-pos. supps(A) ⊇ supps(B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ Converse is very false.

Comparing skew Schur functions quasisymmetrically Peter McNamara 13

Summary so far

sA − sB is Schur-pos. supps(A) ⊇ supps(B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ Converse is very false.

Comparing skew Schur functions quasisymmetrically Peter McNamara 13

Summary so far

sA − sB is Schur-pos. supps(A) ⊇ supps(B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ Converse is very false. Example. A = B = sA = s31 + s211 sB = s22

Comparing skew Schur functions quasisymmetrically Peter McNamara 13

Summary so far

sA − sB is Schur-pos. supps(A) ⊇ supps(B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ Converse is very false. Example. A = B = sA = s31 + s211 sB = s22

New Goal: Find weaker algebraic conditions on A and B that

imply the overlap conditions. What algebraic conditions are being encapsulated by the overlap conditions?

Comparing skew Schur functions quasisymmetrically Peter McNamara 13

Answer: use the F-basis of quasisymmetric functions

Comparing skew Schur functions quasisymmetrically Peter McNamara 14

Answer: use the F-basis of quasisymmetric functions

◮ Skew shape A. ◮ Standard Young tableau (SYT) T of A.

7 5 8 1 4 3 2 6

Comparing skew Schur functions quasisymmetrically Peter McNamara 14

Answer: use the F-basis of quasisymmetric functions

◮ Skew shape A. ◮ Standard Young tableau (SYT) T of A. ◮ Descent set: S(T) = {3, 5}.

7 5 8 1 4 3 2 6

Comparing skew Schur functions quasisymmetrically Peter McNamara 14

Answer: use the F-basis of quasisymmetric functions

◮ Skew shape A. ◮ Standard Young tableau (SYT) T of A. ◮ Descent set: S(T) = {3, 5}.

7 5 8 1 4 3 2 6

Then sA expands in the basis of fundamental quasisymmetric functions as sA =

• SYT T

FS(T). Example. s4431/31 = F{3,5} + · · · .

Comparing skew Schur functions quasisymmetrically Peter McNamara 14

Answer: use the F-basis of quasisymmetric functions

◮ Skew shape A. ◮ Standard Young tableau (SYT) T of A. ◮ Descent set: S(T) = {3, 5}.

7 5 8 1 4 3 2 6

Then sA expands in the basis of fundamental quasisymmetric functions as sA =

• SYT T

FS(T). Example. s4431/31 = F{3,5} + · · · . Facts.

◮ The F form a basis for the quasisymmetric functions. ◮ So notions of F-positivity and F-support make sense. ◮ Schur-positivity implies F-positivity. ◮ supps(A) ⊇ supps(B) implies suppF(A) ⊇ suppF(B)

Comparing skew Schur functions quasisymmetrically Peter McNamara 14

New results: filling the gap

• Theorem. [McN. (2013)]

sA − sB is Schur-pos. sA − sB is F-positive supps(A) ⊇ supps(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ ⇒ ⇓ ⇓

Comparing skew Schur functions quasisymmetrically Peter McNamara 15

New results: filling the gap

• Theorem. [McN. (2013)]

sA − sB is Schur-pos. sA − sB is F-positive supps(A) ⊇ supps(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ ⇒ ⇐ ⇓ ⇓

• Conjecture. The rightmost implication is if and only if.

Comparing skew Schur functions quasisymmetrically Peter McNamara 15

New results: filling the gap

• Theorem. [McN. (2013)]

sA − sB is Schur-pos. sA − sB is F-positive supps(A) ⊇ supps(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ ⇒ ⇐ ⇓ ⇓

• Conjecture. The rightmost implication is if and only if.
• Evidence. Conjecture is true for:

◮ n ≤ 13; ◮ horizontal strips; ◮ F-multiplicity-free skew shapes (as determined by Christine

Bessenrodt and Steph van Willigenburg (2013));

◮ ribbons whose rows all have length at least 2.

Comparing skew Schur functions quasisymmetrically Peter McNamara 15

n = 6 example

F-support containment Dual of row overlap dominance

Comparing skew Schur functions quasisymmetrically Peter McNamara 16

n = 12

n = 12 case has 12,042 edges. n = 13 case has 23,816 edges.

Comparing skew Schur functions quasisymmetrically Peter McNamara 17

Conclusion

sA − sB is Schur-pos. sA − sB is F-positive supps(A) ⊇ supps(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ ⇒ ⇐ ? ⇓ ⇓

Comparing skew Schur functions quasisymmetrically Peter McNamara 18

Conclusion

sA − sB is Schur-pos. sA − sB is F-positive supps(A) ⊇ supps(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ ⇒ ⇐ ? ⇓ ⇓ Thanks! Obrigado!

Comparing skew Schur functions quasisymmetrically Peter McNamara 18

Extras

sA − sB is D-positive sA − sB is Schur-pos. sA − sB is S-positive sA − sB is F-positive suppD(A) ⊇ suppD(B) supps(A) ⊇ supps(B) suppS(A) ⊇ suppS(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ sA − sB is M-positive suppM(A) ⊇ suppM(B)

⇒ ⇒ ⇒ ⇐ ? ⇒ ⇓ ⇓ ⇓ ⇓ ⇓

Comparing skew Schur functions quasisymmetrically Peter McNamara 19

Extras

sA − sB is D-positive sA − sB is Schur-pos. sA − sB is S-positive sA − sB is F-positive suppD(A) ⊇ suppD(B) supps(A) ⊇ supps(B) suppS(A) ⊇ suppS(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ sA − sB is M-positive suppM(A) ⊇ suppM(B)

⇒ ⇒ ⇒ ⇐ ? ⇒ ⇓ ⇓ ⇓ ⇓ ⇓ Conjecture [McN., Alejandro Morales]. A quasisym skew Saturation Theorem: suppF(A) ⊇ suppF(B) ⇐ ⇒ suppF(nA) ⊇ suppF(nB).

Comparing skew Schur functions quasisymmetrically Peter McNamara 19

Extras

sA − sB is D-positive sA − sB is Schur-pos. sA − sB is S-positive sA − sB is F-positive suppD(A) ⊇ suppD(B) supps(A) ⊇ supps(B) suppS(A) ⊇ suppS(B) suppF (A) ⊇ suppF (B) rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ sA − sB is M-positive suppM(A) ⊇ suppM(B)

⇒ ⇒ ⇒ ⇐ ? ⇒ ⇓ ⇓ ⇓ ⇓ ⇓ Conjecture [McN., Alejandro Morales]. A quasisym skew Saturation Theorem: suppF(A) ⊇ suppF(B) ⇐ ⇒ suppF(nA) ⊇ suppF(nB). Thanks! Obrigado!

Comparing skew Schur functions quasisymmetrically Peter McNamara 19