Skew Schur functions: do their row overlaps determine their F - PowerPoint PPT Presentation

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

February 2nd, 2000 F -supports of skew Schur functions Peter McNamara 2

February 2nd, 2000 F -supports of skew Schur functions Peter McNamara 2

February 2nd, 2000 ◮ F -supports of skew Schur functions Peter McNamara 2

February 2nd, 2000 ◮ F -supports of skew Schur functions Peter McNamara 2

February 2nd, 2000 ◮ ◮ 8/28/888 – 2/2/2000 F -supports of skew Schur functions Peter McNamara 2

Preview Conjecture. For skew shapes A and B , supp F ( A ) ⊇ supp F ( B ) ⇐ ⇒ rows k ( A ) � rows k ( B ) for all k . F -supports of skew Schur functions Peter McNamara 3

Preview Conjecture. For skew shapes A and B , supp F ( A ) ⊇ supp F ( B ) ⇐ ⇒ rows k ( A ) � rows k ( B ) for all k . F -supports of skew Schur functions Peter McNamara 3

The beginning of the story s A : the skew Schur function for the skew shape A . Wide Open Question. When is s A = s B ? Determine necessary and sufficient conditions on shapes of A and B . = = F -supports of skew Schur functions Peter McNamara 4

The beginning of the story s A : the skew Schur function for the skew shape A . Wide Open Question. When is s A = s B ? 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

The beginning of the story s A : the skew Schur function for the skew shape A . Wide Open Question. When is s A = s B ? 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

Necessary conditions for equality F -supports of skew Schur functions Peter McNamara 5

Necessary conditions for equality General idea: the overlaps among rows must match up. F -supports of skew Schur functions Peter McNamara 5

Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = F -supports of skew Schur functions Peter McNamara 5

Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. F -supports of skew Schur functions Peter McNamara 5

Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. ◮ overlap 2 ( 1 ) = 2, overlap 2 ( 2 ) = 3, overlap 2 ( 3 ) = 1, overlap 2 ( 4 ) = 1, so rows 2 ( A ) = 3211. F -supports of skew Schur functions Peter McNamara 5

Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. ◮ overlap 2 ( 1 ) = 2, overlap 2 ( 2 ) = 3, overlap 2 ( 3 ) = 1, overlap 2 ( 4 ) = 1, so rows 2 ( A ) = 3211. ◮ rows 3 ( A ) = 11. F -supports of skew Schur functions Peter McNamara 5

Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [RSvW]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. ◮ overlap 2 ( 1 ) = 2, overlap 2 ( 2 ) = 3, overlap 2 ( 3 ) = 1, overlap 2 ( 4 ) = 1, so rows 2 ( A ) = 3211. ◮ rows 3 ( A ) = 11. ◮ rows k ( A ) = ∅ for k > 3. F -supports of skew Schur functions Peter McNamara 5

Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . F -supports of skew Schur functions Peter McNamara 6

Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . supp s ( A ) : Schur support of A supp s ( A ) = { λ : s λ appears in Schur expansion of s A } Example. A = s A = s 3 + 2 s 21 + s 111 supp s ( A ) = { 3 , 21 , 111 } . F -supports of skew Schur functions Peter McNamara 6

Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . supp s ( A ) : Schur support of A supp s ( A ) = { λ : s λ appears in Schur expansion of s A } Example. A = s A = s 3 + 2 s 21 + s 111 supp s ( A ) = { 3 , 21 , 111 } . Theorem [McN., 2008]. Let A and B be skew shapes. If supp s ( A ) = supp s ( B ) , then rows k ( A ) = rows k ( B ) for all k . F -supports of skew Schur functions Peter McNamara 6

Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . supp s ( A ) : Schur support of A supp s ( A ) = { λ : s λ appears in Schur expansion of s A } Example. A = s A = s 3 + 2 s 21 + s 111 supp s ( A ) = { 3 , 21 , 111 } . Theorem [McN., 2008]. Let A and B be skew shapes. If supp s ( A ) = supp s ( B ) , then rows k ( A ) = rows k ( B ) for all k . Converse is definitely not true. F -supports of skew Schur functions Peter McNamara 6

Main interest: inequalities Skew Schur functions are Schur-positive: � c λ s λ/µ = µν s ν . ν Question. What are necessary conditions on A and B if s A − s B is Schur-positive? Theorem [McN., 2008]. Let A and B be skew shapes. If s A − s B is Schur-positive, then rows k ( A ) � rows k ( B ) for all k . F -supports of skew Schur functions Peter McNamara 7

Main interest: inequalities Skew Schur functions are Schur-positive: � c λ s λ/µ = µν s ν . ν Question. What are necessary conditions on A and B if s A − s B is Schur-positive? Theorem [McN., 2008]. Let A and B be skew shapes. If s A − s B is Schur-positive, then rows k ( A ) � rows k ( B ) for all k . In fact, it suffices to assume that supp s ( A ) ⊇ supp s ( B ) . F -supports of skew Schur functions Peter McNamara 7

Summary rows k ( A ) � rows k ( B ) ∀ k Equivalent choices: ⇒ ⇒ s A − s B is Schur-pos. supp s ( A ) ⊇ supp s ( B ) cols ℓ ( A ) � cols ℓ ( B ) ∀ ℓ rects k ,ℓ ( A ) ≤ rects k ,ℓ ( B ) ∀ k , ℓ F -supports of skew Schur functions Peter McNamara 8

Summary rows k ( A ) � rows k ( B ) ∀ k Equivalent choices: ⇒ ⇒ s A − s B is Schur-pos. supp s ( A ) ⊇ supp s ( B ) cols ℓ ( A ) � cols ℓ ( B ) ∀ ℓ rects k ,ℓ ( A ) ≤ rects k ,ℓ ( B ) ∀ k , ℓ Converse is very false. F -supports of skew Schur functions Peter McNamara 8

Summary rows k ( A ) � rows k ( B ) ∀ k Equivalent choices: ⇒ ⇒ s A − s B is Schur-pos. supp s ( A ) ⊇ supp s ( B ) cols ℓ ( A ) � cols ℓ ( B ) ∀ ℓ rects k ,ℓ ( A ) ≤ rects k ,ℓ ( B ) ∀ k , ℓ Converse is very false. Example. A = B = s A = s 31 + s 211 s B = s 22 F -supports of skew Schur functions Peter McNamara 8

Summary rows k ( A ) � rows k ( B ) ∀ k Equivalent choices: ⇒ ⇒ s A − s B is Schur-pos. supp s ( A ) ⊇ supp s ( B ) cols ℓ ( A ) � cols ℓ ( B ) ∀ ℓ rects k ,ℓ ( A ) ≤ rects k ,ℓ ( B ) ∀ k , ℓ Converse is very false. Example. A = B = s A = s 31 + s 211 s B = s 22 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