Comparing skew Schur functions: a quasisymmetric perspective

Transcription

1 Comparing skew Schur functions: a quasisymmetric perspective Peter McNamara Bucknell University AMS/EMS/SPM International Meeting 11 June 2015 Slides and paper available from Comparing skew Schur functions quasisymmetrically Peter McNamara 1

2 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

3 Preview F-support containment Dual of row overlap dominance Comparing skew Schur functions quasisymmetrically Peter McNamara 3

4 Schur functions Cauchy, 1815 Partition λ = (λ 1, λ 2,..., λ l ) Young diagram. Example: λ = (4, 4, 3, 1) Comparing skew Schur functions quasisymmetrically Peter McNamara 4

5 Schur functions Cauchy, 1815 Partition λ = (λ 1, λ 2,..., λ l ) Young diagram. Example: λ = (4, 4, 3, 1) Semistandard Young tableau (SSYT) Comparing skew Schur functions quasisymmetrically Peter McNamara 4

6 Schur functions Cauchy, 1815 Partition λ = (λ 1, λ 2,..., λ l ) Young diagram. Example: λ = (4, 4, 3, 1) Semistandard Young tableau (SSYT) The Schur function s λ in the variables x = (x 1, x 2,...) is then defined by s λ = #1 s in T #2 s in T x 1 x 2. SSYT T Example. s 4431 = x 1 x 2 3 x 4 4 x 5x 2 6 x 7x 9 +. Comparing skew Schur functions quasisymmetrically Peter McNamara 4

7 Skew Schur functions Cauchy, 1815 Partition λ = (λ 1, λ 2,..., λ l ) µ fits inside λ. Young diagram. Example: λ/µ = (4, 4, 3, 1)/(3, 1) Semistandard Young tableau (SSYT) The skew Schur function s λ/µ in the variables x = (x 1, x 2,...) is then defined by s λ/µ = #1 s in T #2 s in T x 1 x 2. SSYT T Example. s 4431/31 = x 3 4 x 5x 2 6 x 7x 9 +. Comparing skew Schur functions quasisymmetrically Peter McNamara 4

8 The beginning of the story s A : the skew Schur function for the skew shape A. Key Facts. s A is symmetric in the variables x 1, x 2,.... The (non-skew) s λ form a basis for the symmetric functions. Comparing skew Schur functions quasisymmetrically Peter McNamara 5

9 The beginning of the story s A : the skew Schur function for the skew shape A. Key Facts. s A is symmetric in the variables x 1, x 2,.... The (non-skew) s λ form a basis for the symmetric functions. Wide Open Question. When is s A = s B? Determine necessary and sufficient conditions on shapes of A and B. = = Comparing skew Schur functions quasisymmetrically Peter McNamara 5

10 The beginning of the story s A : the skew Schur function for the skew shape A. Key Facts. s A is symmetric in the variables x 1, x 2,.... The (non-skew) s λ form a basis for the symmetric functions. 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) Comparing skew Schur functions quasisymmetrically Peter McNamara 5

11 Necessary conditions for equality Comparing skew Schur functions quasisymmetrically Peter McNamara 6

12 Necessary conditions for equality General idea: the overlaps among rows must match up. Comparing skew Schur functions quasisymmetrically Peter McNamara 6

13 Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. 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 = Comparing skew Schur functions quasisymmetrically Peter McNamara 6

14 Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. 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 ith row. Thus rows 1 (A) = Comparing skew Schur functions quasisymmetrically Peter McNamara 6

15 Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. 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 ith row. Thus rows 1 (A) = overlap 2 (1) = 2, overlap 2 (2) = 3, overlap 2 (3) = 1, overlap 2 (4) = 1, so rows 2 (A) = Comparing skew Schur functions quasisymmetrically Peter McNamara 6

16 Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. 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 ith row. Thus rows 1 (A) = overlap 2 (1) = 2, overlap 2 (2) = 3, overlap 2 (3) = 1, overlap 2 (4) = 1, so rows 2 (A) = rows 3 (A) = 11. Comparing skew Schur functions quasisymmetrically Peter McNamara 6

17 Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. 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 ith row. Thus rows 1 (A) = overlap 2 (1) = 2, overlap 2 (2) = 3, overlap 2 (3) = 1, overlap 2 (4) = 1, so rows 2 (A) = rows 3 (A) = 11. rows k (A) = for k > 3. Comparing skew Schur functions quasisymmetrically Peter McNamara 6

18 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. Comparing skew Schur functions quasisymmetrically Peter McNamara 7

19 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 + 2s 21 + s 111 supp s (A) = {3, 21, 111}. Comparing skew Schur functions quasisymmetrically Peter McNamara 7

20 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 + 2s 21 + s 111 supp s (A) = {3, 21, 111}. Theorem [McN., 2008]. It suffices to assume supp s (A) = supp s (B). Comparing skew Schur functions quasisymmetrically Peter McNamara 7

21 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 + 2s 21 + s 111 supp s (A) = {3, 21, 111}. Theorem [McN., 2008]. It suffices to assume supp s (A) = supp s (B). Converse is definitely not true: Comparing skew Schur functions quasisymmetrically Peter McNamara 7

22 Schur-positivity order Our interest: inequalities. Skew Schur functions are Schur-positive: s λ/µ = ν c λ µνs ν. Original Question. When is s λ/µ s σ/τ Schur-positive? Comparing skew Schur functions quasisymmetrically Peter McNamara 8

23 Schur-positivity order Our interest: inequalities. Skew Schur functions are Schur-positive: s λ/µ = ν c λ µνs ν. Original Question. When is s λ/µ s σ/τ Schur-positive? Definition. Let A, B be skew shapes. We say that A s B if s A s B is Schur-positive. Original goal: Characterize the Schur-positivity order s in terms of skew shapes. Comparing skew Schur functions quasisymmetrically Peter McNamara 8

24 Example of a Schur-positivity poset If B s A then A = B. Call the resulting ordered set P n. Then P 4 : Comparing skew Schur functions quasisymmetrically Peter McNamara 9

25 More examples P 5 : P 6 : Comparing skew Schur functions quasisymmetrically Peter McNamara 10

26 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

27 Necessary conditions for Schur-positivity Comparing skew Schur functions quasisymmetrically Peter McNamara 12

28 Necessary conditions for Schur-positivity Notation. Write λ µ if λ is less than or equal to µ in dominance order, i.e. λ 1 + λ i µ 1 + µ i for all i. Comparing skew Schur functions quasisymmetrically Peter McNamara 12

29 Necessary conditions for Schur-positivity Notation. Write λ µ if λ is less than or equal to µ in dominance order, i.e. λ 1 + λ i µ 1 + µ i for all i. 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. Comparing skew Schur functions quasisymmetrically Peter McNamara 12

30 Necessary conditions for Schur-positivity Notation. Write λ µ if λ is less than or equal to µ in dominance order, i.e. λ 1 + λ i µ 1 + µ i for all i. 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). Comparing skew Schur functions quasisymmetrically Peter McNamara 12

31 Necessary conditions for Schur-positivity Notation. Write λ µ if λ is less than or equal to µ in dominance order, i.e. λ 1 + λ i µ 1 + µ i for all i. 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). Application. A = B = rows 1 (A) = 3221 rows 1 (B) = 2221 Comparing skew Schur functions quasisymmetrically Peter McNamara 12

32 Necessary conditions for Schur-positivity Notation. Write λ µ if λ is less than or equal to µ in dominance order, i.e. λ 1 + λ i µ 1 + µ i for all i. 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). Application. A = B = rows 1 (A) = 3221 rows 1 (B) = 2221 rows 2 (B) = 21 rows 2 (A) = 111 Comparing skew Schur functions quasisymmetrically Peter McNamara 12

33 Necessary conditions for Schur-positivity Notation. Write λ µ if λ is less than or equal to µ in dominance order, i.e. λ 1 + λ i µ 1 + µ i for all i. 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). Application. A = B = rows 1 (A) = 3221 rows 1 (B) = 2221 rows 2 (B) = 21 rows 2 (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

34 Summary so far s A s B is Schur-pos. supp s (A) supp s (B) rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l Comparing skew Schur functions quasisymmetrically Peter McNamara 13

35 Summary so far s A s B is Schur-pos. supp s (A) supp s (B) rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l Converse is very false. Comparing skew Schur functions quasisymmetrically Peter McNamara 13

36 Summary so far s A s B is Schur-pos. supp s (A) supp s (B) rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l Converse is very false. Comparing skew Schur functions quasisymmetrically Peter McNamara 13

37 Summary so far s A s B is Schur-pos. supp s (A) supp s (B) rows k (A) rows k (B) k 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 Comparing skew Schur functions quasisymmetrically Peter McNamara 13

38 Summary so far s A s B is Schur-pos. supp s (A) supp s (B) rows k (A) rows k (B) k 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 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

39 Answer: use the F -basis of quasisymmetric functions Comparing skew Schur functions quasisymmetrically Peter McNamara 14

40 Answer: use the F -basis of quasisymmetric functions Skew shape A. Standard Young tableau (SYT) T of A Comparing skew Schur functions quasisymmetrically Peter McNamara 14

41 Answer: use the F -basis of quasisymmetric functions Skew shape A. Standard Young tableau (SYT) T of A. Descent set: S(T ) = {3, 5} Comparing skew Schur functions quasisymmetrically Peter McNamara 14

42 Answer: use the F -basis of quasisymmetric functions Skew shape A. Standard Young tableau (SYT) T of A. Descent set: S(T ) = {3, 5}. 6 7 Then s A expands in the basis of fundamental quasisymmetric functions as s A = F S(T ). Example. SYT T s 4431/31 = F {3,5} Comparing skew Schur functions quasisymmetrically Peter McNamara 14

43 Answer: use the F -basis of quasisymmetric functions Skew shape A. Standard Young tableau (SYT) T of A. Descent set: S(T ) = {3, 5}. 6 7 Then s A expands in the basis of fundamental quasisymmetric functions as s A = F S(T ). Example. Facts. SYT T s 4431/31 = F {3,5} +. The F form a basis for the quasisymmetric functions. So notions of F-positivity and F-support make sense. Schur-positivity implies F-positivity. supp s (A) supp s (B) implies supp F (A) supp F (B) Comparing skew Schur functions quasisymmetrically Peter McNamara

44 New results: filling the gap Theorem. [McN. (2013)] s A s B is Schur-pos. supp s (A) supp s (B) s A s B is F -positive supp F (A) supp F (B) rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l Comparing skew Schur functions quasisymmetrically Peter McNamara 15

45 New results: filling the gap Theorem. [McN. (2013)] s A s B is Schur-pos. supp s (A) supp s (B) s A s B is F -positive supp F (A) supp F (B) rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l Conjecture. The rightmost implication is if and only if. Comparing skew Schur functions quasisymmetrically Peter McNamara 15

46 New results: filling the gap Theorem. [McN. (2013)] s A s B is Schur-pos. supp s (A) supp s (B) s A s B is F -positive supp F (A) supp F (B) rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,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

47 n = 6 example F-support containment Dual of row overlap dominance Comparing skew Schur functions quasisymmetrically Peter McNamara 16

48 n = 12 n = 12 case has 12,042 edges. n = 13 case has 23,816 edges. Comparing skew Schur functions quasisymmetrically Peter McNamara 17

49 Conclusion s A s B is Schur-pos. supp s (A) supp s (B) s A s B is F -positive supp F (A) supp F (B)? rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l Comparing skew Schur functions quasisymmetrically Peter McNamara 18

50 Conclusion s A s B is Schur-pos. supp s (A) supp s (B) s A s B is F -positive supp F (A) supp F (B)? rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l Thanks! Obrigado! Comparing skew Schur functions quasisymmetrically Peter McNamara 18

51 Extras s A s B is D-positive s A s B is Schur-pos. s A s B is S-positive supp D (A) supp D (B) supp s (A) supp s (B) supp S (A) supp S (B) s A s B is F -positive supp F (A) supp F (B)? rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l s A s B is M-positive supp M (A) supp M (B) Comparing skew Schur functions quasisymmetrically Peter McNamara 19

52 Extras s A s B is D-positive s A s B is Schur-pos. s A s B is S-positive supp D (A) supp D (B) supp s (A) supp s (B) supp S (A) supp S (B) s A s B is F -positive supp F (A) supp F (B)? rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l s A s B is M-positive supp M (A) supp M (B) Conjecture [McN., Alejandro Morales]. A quasisym skew Saturation Theorem: supp F (A) supp F (B) supp F (na) supp F (nb). Comparing skew Schur functions quasisymmetrically Peter McNamara 19

53 Extras s A s B is D-positive s A s B is Schur-pos. s A s B is S-positive supp D (A) supp D (B) supp s (A) supp s (B) supp S (A) supp S (B) s A s B is F -positive supp F (A) supp F (B)? rows k (A) rows k (B) k cols l (A) cols l (B) l rects k,l (A) rects k,l (B) k, l s A s B is M-positive supp M (A) supp M (B) Conjecture [McN., Alejandro Morales]. A quasisym skew Saturation Theorem: supp F (A) supp F (B) supp F (na) supp F (nb). Thanks! Obrigado! Comparing skew Schur functions quasisymmetrically Peter McNamara 19