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Skew Schur functions: do their row overlaps determine their F -supports? Peter McNamara Bucknell University Stanley@70 24 June 2014 Slides and paper available from www.facstaff.bucknell.edu/pm040/ F -supports of skew Schur functions Peter

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Skew Schur functions: do their row overlaps determine their F-supports?

Peter McNamara Bucknell University Stanley@70 24 June 2014 Slides and paper available from www.facstaff.bucknell.edu/pm040/

F-supports of skew Schur functions Peter McNamara 1

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February 2nd, 2000

F-supports of skew Schur functions Peter McNamara 2

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February 2nd, 2000

F-supports of skew Schur functions Peter McNamara 2

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February 2nd, 2000

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F-supports of skew Schur functions Peter McNamara 2

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February 2nd, 2000

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F-supports of skew Schur functions Peter McNamara 2

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February 2nd, 2000

◮ ◮ 8/28/888 – 2/2/2000

F-supports of skew Schur functions Peter McNamara 2

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Preview

  • Conjecture. For skew shapes A and B,

suppF(A) ⊇ suppF(B) ⇐ ⇒ rowsk(A) rowsk(B) for all k.

F-supports of skew Schur functions Peter McNamara 3

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Preview

  • Conjecture. For skew shapes A and B,

suppF(A) ⊇ suppF(B) ⇐ ⇒ rowsk(A) rowsk(B) for all k.

F-supports of skew Schur functions Peter McNamara 3

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The beginning of the story

sA: the skew Schur function for the skew shape A. Wide Open Question. When is sA = sB? Determine necessary and sufficient conditions on shapes of A and B. = =

F-supports of skew Schur functions Peter McNamara 4

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The beginning of the story

sA: the skew Schur function for the skew shape A. 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)

F-supports of skew Schur functions Peter McNamara 4

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The beginning of the story

sA: the skew Schur function for the skew shape A. 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)

But this is not the problem I want to talk about....

F-supports of skew Schur functions Peter McNamara 4

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Necessary conditions for equality

F-supports of skew Schur functions Peter McNamara 5

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Necessary conditions for equality

General idea: the overlaps among rows must match up.

F-supports of skew Schur functions Peter McNamara 5

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Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A, let overlapk(i) be the number

  • f 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 =

F-supports of skew Schur functions Peter McNamara 5

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Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A, let overlapk(i) be the number

  • f 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.

F-supports of skew Schur functions Peter McNamara 5

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Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A, let overlapk(i) be the number

  • f 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.

F-supports of skew Schur functions Peter McNamara 5

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Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A, let overlapk(i) be the number

  • f 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.

F-supports of skew Schur functions Peter McNamara 5

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Necessary conditions for equality

General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A, let overlapk(i) be the number

  • f 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.

F-supports of skew Schur functions Peter McNamara 5

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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.

F-supports of skew Schur functions Peter McNamara 6

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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}.

F-supports of skew Schur functions Peter McNamara 6

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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]. Let A and B be skew shapes. If supps(A) = supps(B), then rowsk(A) = rowsk(B) for all k.

F-supports of skew Schur functions Peter McNamara 6

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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]. Let A and B be skew shapes. If supps(A) = supps(B), then rowsk(A) = rowsk(B) for all k. Converse is definitely not true.

F-supports of skew Schur functions Peter McNamara 6

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Main interest: inequalities

Skew Schur functions are Schur-positive: sλ/µ =

  • ν

cλ

µνsν.

  • Question. What are necessary conditions on A and B if sA − sB is

Schur-positive? Theorem [McN., 2008]. Let A and B be skew shapes. If sA − sB is Schur-positive, then rowsk(A) rowsk(B) for all k.

F-supports of skew Schur functions Peter McNamara 7

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Main interest: inequalities

Skew Schur functions are Schur-positive: sλ/µ =

  • ν

cλ

µνsν.

  • Question. What are necessary conditions on A and B if sA − sB is

Schur-positive? 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).

F-supports of skew Schur functions Peter McNamara 7

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Summary

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

⇒ ⇒

F-supports of skew Schur functions Peter McNamara 8

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Summary

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

⇒ ⇒ Converse is very false.

F-supports of skew Schur functions Peter McNamara 8

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Summary

sA − sB is Schur-pos. supps(A) ⊇ supps(B) rowsk(A) rowsk(B) ∀k Equivalent choices: 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

F-supports of skew Schur functions Peter McNamara 8

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Summary

sA − sB is Schur-pos. supps(A) ⊇ supps(B) rowsk(A) rowsk(B) ∀k Equivalent choices: 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

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

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

F-supports of skew Schur functions Peter McNamara 8

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The quasisymmetric perspective

Theorem [Gessel & Stanley]. sA: nice expansion in Gessel’s fundamental quasisymmetric basis F. Theorem [McN., 2013].

sA − sB is Schur-pos. sA − sB is F-positive supps(A) ⊇ supps(B) suppF

F F (A) ⊇ suppF F F (B)

rowsk(A) rowsk(B) ∀k colsℓ(A) colsℓ(B) ∀ℓ rectsk,ℓ(A) ≤ rectsk,ℓ(B) ∀k, ℓ

⇒ ⇒ ⇒ ⇓ ⇓

F-supports of skew Schur functions Peter McNamara 9

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The quasisymmetric perspective

Theorem [Gessel & Stanley]. sA: nice expansion in Gessel’s fundamental quasisymmetric basis F. Theorem [McN., 2013].

sA − sB is Schur-pos. sA − sB is F-positive supps(A) ⊇ supps(B) suppF

F F (A) ⊇ suppF F F (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.

F-supports of skew Schur functions Peter McNamara 9

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n = 6 example

F-support containment Dual of row overlap dominance

F-supports of skew Schur functions Peter McNamara 10

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n = 12 case has 12,042 edges

F-supports of skew Schur functions Peter McNamara 11

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n = 12 case has 12,042 edges

Conjecture.

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

⇒ ⇒ ⇒ ⇐ ? ⇓ ⇓

F-supports of skew Schur functions Peter McNamara 11

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n = 12 case has 12,042 edges

Conjecture.

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

F-supports of skew Schur functions Peter McNamara 11

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Adding other bases

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)

⇒ ⇒ ⇒ ⇐ ? ⇒ ⇓ ⇓ ⇓ ⇓ ⇓

F-supports of skew Schur functions Peter McNamara 12