Conjectures concerning the difference of two skew Schur functions

Transcription

1 Conjectures concerning the difference of two skew Schur functions Peter McNamara Bucknell University Positivity in Algebraic Combinatorics Banff International Research Station 15 August 2015 Slides and papers available from Conjectures on differences of skew Schurs Peter McNamara 1

2 The setting s A : the skew Schur function for the skew shape A Overarching Question. For skew shapes A and B, when is s A s B Schur-positive? Want simple conditions in terms of the shapes of A and B. Conjectures on differences of skew Schurs Peter McNamara 2

3 The setting s A : the skew Schur function for the skew shape A Overarching Question. For skew shapes A and B, when is s A s B Schur-positive? Want simple conditions in terms of the shapes of A and B. Special Case. For partitions α, β, γ, δ, when is Schur-positive? s α s β s γ s δ = [Azenhas, Ballantine, F. Bergeron, Biagioli, Conflitti, Fomin, Fulton, King, A. N. Kirillov, Lam, Lascoux, Leclerc, C.-K. Li, Mamede, M., Okounkov, Orellana, Poon, Postnikov, Pylyavskyy, Rosas, Thibon, Welsh, van Willigenburg,...] Conjectures on differences of skew Schurs Peter McNamara 2

4 The problems and conjectures 1. Equality of skew Schur functions Joint with Stephanie van Willigenburg 2. Connected skew Schur functions maximal in Schur-positivity order Joint with Pavlo Pylyavskyy and Stephanie van Willigenburg 3. F -support containment and the row-overlap conditions of Reiner, Shaw and van Willigenburg 4. A Saturation Theorem for skew Schur functions Joint with Alejandro Morales Conjectures on differences of skew Schurs Peter McNamara 3

5 The problems and conjectures 1. Equality of skew Schur functions Joint with Stephanie van Willigenburg 2. Connected skew Schur functions maximal in Schur-positivity order Joint with Pavlo Pylyavskyy and Stephanie van Willigenburg 3. F -support containment and the row-overlap conditions of Reiner, Shaw and van Willigenburg 4. A Saturation Theorem for skew Schur functions Joint with Alejandro Morales Conjectures on differences of skew Schurs Peter McNamara 3

6 1. Equality of skew Schur functions Conjectures on differences of skew Schurs Peter McNamara 4

7 1. Equality of skew Schur functions Problem 1. When is s A = s B? Denoted A B. Determine necessary and sufficient conditions on shapes of A and B. Conjectures on differences of skew Schurs Peter McNamara 4

8 1. Equality of skew Schur functions Problem 1. When is s A = s B? Denoted A B. Determine necessary and sufficient conditions on shapes of A and B. Lou Billera, Hugh Thomas, Steph van Willigenburg (2004): complete answer for ribbons Conjectures on differences of skew Schurs Peter McNamara 4

9 1. Equality of skew Schur functions Problem 1. When is s A = s B? Denoted A B. Determine necessary and sufficient conditions on shapes of A and B. Lou Billera, Hugh Thomas, Steph van Willigenburg (2004): complete answer for ribbons John Stembridge (2004): skewed staircases Conjectures on differences of skew Schurs Peter McNamara 4

10 1. Equality of skew Schur functions Problem 1. When is s A = s B? Denoted A B. Determine necessary and sufficient conditions on shapes of A and B. Lou Billera, Hugh Thomas, Steph van Willigenburg (2004): complete answer for ribbons John Stembridge (2004): skewed staircases Vic Reiner, Kristin Shaw, Steph van Willigenburg (2006): 3 operations for generating skew shapes with equal skew Schur functions; necessary conditions M., Steph van Willigenburg (2006): unification, generalization, conjecture for necessary and sufficient conditions Christian Gutschwager (2008): multiplicity-free skew shapes Conjectures on differences of skew Schurs Peter McNamara 4

11 1. Equality of skew Schur functions (With apologies) Conjecture 1 [M., van Willigenburg (2006); inspired by main result of BTvW (2006)]. Two skew shapes E and E satisfy E E if and only if, for some r, E = (( (E 1 W2 E 2 ) W3 E 3 ) ) Wr E r E = (( (E 1 W E 2 2 ) W E 3 3 ) ) W r E r, where E i = W i O i W i satisfies four hypotheses for all i, E and W denote either E i i i and W i, or E and W. i i Conjectures on differences of skew Schurs Peter McNamara 5

12 1. Equality of skew Schur functions (With apologies) Conjecture 1 [M., van Willigenburg (2006); inspired by main result of BTvW (2006)]. Two skew shapes E and E satisfy E E if and only if, for some r, E = (( (E 1 W2 E 2 ) W3 E 3 ) ) Wr E r E = (( (E 1 W E 2 2 ) W E 3 3 ) ) W r E r, where E i = W i O i W i satisfies four hypotheses for all i, E and W denote either E i i i and W i, or E and W. i i Evidence [M., van Willigenburg, (2006)]. With one more hypothesis, the if direction Proof uses results of Hamel Goulden and Chen Yan Yang. n 20 Evidence [Gutschwager, 2006)]. Multiplicity-free skew shapes Conjectures on differences of skew Schurs Peter McNamara 5

13 2. Maximal connected skew shapes Conjectures on differences of skew Schurs Peter McNamara 6

14 2. Maximal connected skew shapes Definition. Let A, B be skew shapes. We say that A s B if s A s B is Schur-positive. If B s A then A = B. Example. P 4 : Conjectures on differences of skew Schurs Peter McNamara 6

15 2. Maximal connected skew shapes Definition. Let A, B be skew shapes. We say that A s B if s A s B is Schur-positive. If B s A then A = B. Example. P 4 : Conjectures on differences of skew Schurs Peter McNamara 6

16 2. Maximal connected skew shapes Definition. Let A, B be skew shapes. We say that A s B if s A s B is Schur-positive. If B s A then A = B. Example. P 4 : Problem 2. What are the maximal elements of P n among the connected skew shapes? Conjectures on differences of skew Schurs Peter McNamara 6

17 2. Maximal connected skew shapes Conjecture 2 [M., Pylyavskyy (2007)]. For each r = 1,..., n, there is a unique maximal connected element with r rows, namely the ribbon marked out by the diagonal of an r-by-(n r + 1) box. Examples. Conjectures on differences of skew Schurs Peter McNamara 7

18 2. Maximal connected skew shapes Conjecture 2 [M., Pylyavskyy (2007)]. For each r = 1,..., n, there is a unique maximal connected element with r rows, namely the ribbon marked out by the diagonal of an r-by-(n r + 1) box. Examples. Evidence [M., van Willigenburg (2011)]. n 34 Maximal element must be an equitable ribbon: row (resp. column) lengths differ by at most 1. Supp s (A) {λ n s λ appears in the Schur expansion of s A }, the Schur-support of A. e.g. s = s 3 + 2s 21 + s 111. Supp s ( ) = {3, 21, 111}. True in Support Poset: A Supps B if Supp s (A) Supp s (B). Conjectures on differences of skew Schurs Peter McNamara 7

19 3. The row-overlap conditions Conjectures on differences of skew Schurs Peter McNamara 8

20 3. The row-overlap conditions General idea: the overlaps among rows must match up for s A = s B. Conjectures on differences of skew Schurs Peter McNamara 8

21 3. The row-overlap conditions General idea: the overlaps among rows must match up for s A = s B. 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 = Conjectures on differences of skew Schurs Peter McNamara 8

22 3. The row-overlap conditions General idea: the overlaps among rows must match up for s A = s B. 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) = Conjectures on differences of skew Schurs Peter McNamara 8

23 3. The row-overlap conditions General idea: the overlaps among rows must match up for s A = s B. 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) = Conjectures on differences of skew Schurs Peter McNamara 8

24 3. The row-overlap conditions General idea: the overlaps among rows must match up for s A = s B. 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. Conjectures on differences of skew Schurs Peter McNamara 8

25 3. The row-overlap conditions General idea: the overlaps among rows must match up for s A = s B. 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. Conjectures on differences of skew Schurs Peter McNamara 8

26 3. The row-overlap conditions Necessary conditons 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. Question. What are necessary conditions on A and B for s A s B to be Schur-positive? Theorem [M., (2008)]. Let A and B be skew shapes. If s A s B is Schur-positive, then rows k (A) dom rows k (B) for all k. Conjectures on differences of skew Schurs Peter McNamara 9

27 3. The row-overlap conditions Necessary conditons 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. Question. What are necessary conditions on A and B for s A s B to be Schur-positive? Theorem [M., (2008)]. Let A and B be skew shapes. If s A s B is Schur-positive, then rows k (A) dom rows k (B) for all k. In fact, it suffices to assume that Supp s (A) Supp s (B). Conjectures on differences of skew Schurs Peter McNamara 9

28 3. The row-overlap conditions Theorem [M., (2008)]. s A s B is Schur-pos. Supp s (A) Supp s (B) rows k (A) dom rows k (B) k Equivalent choices: cols l (A) dom cols l (B) l rects k,l (A) rects k,l (B) k, l Conjectures on differences of skew Schurs Peter McNamara 10

29 3. The row-overlap conditions Theorem [M., (2008)]. s A s B is Schur-pos. Supp s (A) Supp s (B) Converse is already false at n = 4. rows k (A) dom rows k (B) k Equivalent choices: cols l (A) dom cols l (B) l rects k,l (A) rects k,l (B) k, l Problem 3. What weaker algebraic conditions best fill the gap? Conjectures on differences of skew Schurs Peter McNamara 10

30 3. The row-overlap conditions and F-support containment Theorem [M., (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) dom rows k (B) k Equivalent choices: cols l (A) dom cols l (B) l rects k,l (A) rects k,l (B) k, l Problem 3. What weaker algebraic conditions best fill the gap? Conjectures on differences of skew Schurs Peter McNamara 10

31 3. The row-overlap conditions and F-support containment Theorem [M., (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) dom rows k (B) k Equivalent choices: cols l (A) dom cols l (B) l rects k,l (A) rects k,l (B) k, l Problem 3. What weaker algebraic conditions best fill the gap? Conjecture 4 [M., (2013)]. The rightmost implication is if and only if. Conjectures on differences of skew Schurs Peter McNamara 10

32 3. The row-overlap conditions and F-support containment Theorem [M., (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) dom rows k (B) k Equivalent choices: cols l (A) dom cols l (B) l rects k,l (A) rects k,l (B) k, l Problem 3. What weaker algebraic conditions best fill the gap? Conjecture 4 [M., (2013)]. The rightmost implication is if and only if. Evidence [M., (2013)]. Conjecture is true for: n 13 (compare with failure at n = 4 for other converse implications); F-multiplicity-free skew shapes (as classified by Christine Bessenrodt and Steph van Willigenburg, (2013)); ribbons whose rows all have length at least 2. Conjectures on differences of skew Schurs Peter McNamara 10

33 3. The row-overlap conditions and F-support containment Example. n = 6 F-support containment Dual of row overlap dominance Conjectures on differences of skew Schurs Peter McNamara 11

34 3. The row-overlap conditions and F-support containment Example. n = 12 case has 12,042 edges. Conjectures on differences of skew Schurs Peter McNamara 12

35 4. A Saturation Theorem for skew Schur functions Conjectures on differences of skew Schurs Peter McNamara 13

36 4. A Saturation Theorem for skew Schur functions A = λ/µ and k is a positive integer, define ka = kλ/kµ. Theorem [Knutson, Tao, (1999)]. For a skew shape A and partition ν, ν Supp s (A) nν Supp s (na). Conjectures on differences of skew Schurs Peter McNamara 13

37 4. A Saturation Theorem for skew Schur functions A = λ/µ and k is a positive integer, define ka = kλ/kµ. Theorem [Knutson, Tao, (1999)]. For a skew shape A and partition ν, ν Supp s (A) nν Supp s (na). Equivalently, Supp s (ν) Supp s (A) Supp s (nν) Supp s (na). Conjectures on differences of skew Schurs Peter McNamara 13

38 4. A Saturation Theorem for skew Schur functions A = λ/µ and k is a positive integer, define ka = kλ/kµ. Theorem [Knutson, Tao, (1999)]. For a skew shape A and partition ν, ν Supp s (A) nν Supp s (na). Equivalently, Supp s (ν) Supp s (A) Supp s (nν) Supp s (na). Problem 4. [Speyer (2009)]. Can this be generalized by replacing ν by a skew shape? Conjectures on differences of skew Schurs Peter McNamara 13

39 4. A Saturation Theorem for skew Schur functions A = λ/µ and k is a positive integer, define ka = kλ/kµ. Theorem [Knutson, Tao, (1999)]. For a skew shape A and partition ν, ν Supp s (A) nν Supp s (na). Equivalently, Supp s (ν) Supp s (A) Supp s (nν) Supp s (na). Problem 4. [Speyer (2009)]. Can this be generalized by replacing ν by a skew shape? Answer. No. False even in the easier direction. Conjectures on differences of skew Schurs Peter McNamara 13

40 4. A Saturation Theorem for skew Schur functions A = λ/µ and k is a positive integer, define ka = kλ/kµ. Theorem [Knutson, Tao, (1999)]. For a skew shape A and partition ν, ν Supp s (A) nν Supp s (na). Equivalently, Supp s (ν) Supp s (A) Supp s (nν) Supp s (na). Problem 4. [Speyer (2009)]. Can this be generalized by replacing ν by a skew shape? Answer. No. False even in the easier direction. Conjecture 4 [M., Morales (2014)]. A quasisymmetric skew Saturation Theorem: Supp F (B) Supp F (A) Supp F (nb) Supp F (na). Conjectures on differences of skew Schurs Peter McNamara 13

41 4. A Saturation Theorem for skew Schur functions A = λ/µ and k is a positive integer, define ka = kλ/kµ. Theorem [Knutson, Tao, (1999)]. For a skew shape A and partition ν, ν Supp s (A) nν Supp s (na). Equivalently, Supp s (ν) Supp s (A) Supp s (nν) Supp s (na). Problem 4. [Speyer (2009)]. Can this be generalized by replacing ν by a skew shape? Answer. No. False even in the easier direction. Conjecture 4 [M., Morales (2014)]. A quasisymmetric skew Saturation Theorem: Supp F (B) Supp F (A) Supp F (nb) Supp F (na). Evidence. Follows from Conjecture 3. Conjectures on differences of skew Schurs Peter McNamara 13

42 1. Equality of skew Schur functions Conjectures on differences of skew Schurs Peter McNamara 14

43 1. Equality of skew Schur functions Conjecture 1 [M., van Willigenburg (2006); inspired by main result of BTvW (2006)]. Two skew shapes E and E satisfy E E if and only if, for some r, E = (( (E 1 W2 E 2 ) W3 E 3 ) ) Wr E r E = (( (E 1 W E 2 2 ) W E 3 3 ) ) W r E r, where E i = W i O i W i satisfies four hypotheses for all i, E and W denote either E i i i and W i, or E and W. i i Conjectures on differences of skew Schurs Peter McNamara 14

44 1. Equality of skew Schur functions Composition of skew shapes D E = Conjectures on differences of skew Schurs Peter McNamara 15

45 1. Equality of skew Schur functions Composition of skew shapes D E = Conjectures on differences of skew Schurs Peter McNamara 15

46 1. Equality of skew Schur functions Composition of skew shapes D E = Conjectures on differences of skew Schurs Peter McNamara 15

47 1. Equality of skew Schur functions Composition of skew shapes D E = Conjectures on differences of skew Schurs Peter McNamara 15

48 1. Equality of skew Schur functions Composition of skew shapes D E = Theorem [M., van Willigenburg, (2006)]. If D D, then D E D E D E. Conjectures on differences of skew Schurs Peter McNamara 15

49 1. Equality of skew Schur functions Amalgamated compositions: W A skew shape W lies in the top of a skew shape E if W appears as a connected subshape of E that includes the northeasternmost cell of E. W W W W W Conjectures on differences of skew Schurs Peter McNamara 16

50 1. Equality of skew Schur functions Amalgamated compositions: W A skew shape W lies in the top of a skew shape E if W appears as a connected subshape of E that includes the northeasternmost cell of E. W W W W W Similarly, W lies in the bottom of E. W W W W W Our interest. W lies in both the top and bottom of E. We write E = WOW. Conjectures on differences of skew Schurs Peter McNamara 16

51 1. Equality of skew Schur functions Amalgamated compositions: W A skew shape W lies in the top of a skew shape E if W appears as a connected subshape of E that includes the northeasternmost cell of E. W W W W W Similarly, W lies in the bottom of E. W W W W W Our interest. W lies in both the top and bottom of E. We write E = WOW. Hypotheses [inspired by hypotheses of RSvW]. 1. W ne and W sw are separated by at least one diagonal. 2. E \ W ne and E \ W sw are both connected skew shapes. 3. W is maximal given its set of diagonals. Conjectures on differences of skew Schurs Peter McNamara 16

52 1. Equality of skew Schur functions Example. D W E = Conjectures on differences of skew Schurs Peter McNamara 17

53 1. Equality of skew Schur functions Example. D W E = Conjectures on differences of skew Schurs Peter McNamara 17

54 1. Equality of skew Schur functions Example. D W E = Conjectures on differences of skew Schurs Peter McNamara 17

55 1. Equality of skew Schur functions Example. D W E = Conjectures on differences of skew Schurs Peter McNamara 17

56 1. Equality of skew Schur functions Example. D W E = Conjectures on differences of skew Schurs Peter McNamara 17

57 1. Equality of skew Schur functions Construction of W and O: W W W W W W W W W W W W W W W Conjectures on differences of skew Schurs Peter McNamara 18

58 1. Equality of skew Schur functions Construction of W and O: W W W W W W W W W W W W W W W Conjectures on differences of skew Schurs Peter McNamara 18

59 1. Equality of skew Schur functions Construction of W and O: W W W W W W W W W W Hypothesis 4. W is never adjacent to O. W W W W W Conjectures on differences of skew Schurs Peter McNamara 18

60 1. Equality of skew Schur functions Construction of W and O: W W W W W W W W W W Hypothesis 4. W is never adjacent to O. W W W W W Conjecture 1. Two skew shapes E and E satisfy E E if and only if, for some r, E = (( (E 1 W2 E 2 ) W3 E 3 ) ) Wr E r E = (( (E 1 W E 2 2 ) W E 3 3 ) ) W r E r, where E i = W i O i W i satisfies Hypotheses 1 4 for all i, E and W denote either E i i i and W i, or E and W. i i Conjectures on differences of skew Schurs Peter McNamara 18

61 1. Equality of skew Schur functions Construction of W and O: W W W W W W W W W W Hypothesis 4. W is never adjacent to O. W W W W W Conjecture 1. Two skew shapes E and E satisfy E E if and only if, for some r, E = (( (E 1 W2 E 2 ) W3 E 3 ) ) Wr E r E = (( (E 1 W E 2 2 ) W E 3 3 ) ) W r E r, where E i = W i O i W i satisfies Hypotheses 1 4 for all i, E and W denote either E i i i and W i, or E and W. i i Thanks! Conjectures on differences of skew Schurs Peter McNamara 18