Resonance in orbits of plane partitions and increasing tableaux. Jessica Striker North Dakota State University

Transcription

1 Resonance in orbits of plane partitions and increasing tableaux Jessica Striker North Dakota State University joint work with Kevin Dilks (NDSU) and Oliver Pechenik (UIUC) February 19, 2016 J. Striker (NDSU) Resonance February 19, / 69

2 Resonance in orbits of plane partitions and increasing tableaux 1 Resonance defined 2 Resonance of rowmotion 3 Multidimensional promotion and rowmotion 4 Resonance of K-promotion 5 A bijection between increasing tableaux and plane partitions J. Striker (NDSU) Resonance February 19, / 69

3 Resonance in orbits of plane partitions and increasing tableaux 1 Resonance defined 2 Resonance of rowmotion 3 Multidimensional promotion and rowmotion 4 Resonance of K-promotion 5 A bijection between increasing tableaux and plane partitions J. Striker (NDSU) Resonance February 19, / 69

4 Resonance in musical instruments J. Striker (NDSU) Resonance February 19, / 69

5 Resonance in dynamical algebraic combinatorics J. Striker (NDSU) Resonance February 19, / 69

6 Resonance in dynamical algebraic combinatorics Definition Dynamical algebraic combinatorics is the study of actions on objects important in algebraic combinatorics. Compose local actions to get interesting global actions. Toy example: (12)(23)(34)(45)(56) = (123456) Examples of interesting actions in algebraic combinatorics: Gyration on alternating sign matrices Promotion on tableaux Rowmotion on order ideals of a poset J. Striker (NDSU) Resonance February 19, / 69

7 Resonance in dynamical algebraic combinatorics Definition (Dilks, Pechenik, Striker 2015+) Let G = g be a cyclic group acting on a set X, C ω = c a cyclic group of order ω acting nontrivially on a set Y, and f : X Y a surjection. If c f (x) = f (g x) for all x X, we say the triple (X, G, f ) exhibits resonance with frequency ω. X g X f f Y c Y J. Striker (NDSU) Resonance February 19, / 69

8 Alternating sign matrix definition Definition Alternating sign matrices (ASMs) are square matrices with the following properties: entries {0, 1, 1} each row and each column sums to 1 nonzero entries alternate in sign along a row/column J. Striker (NDSU) Resonance February 19, / 69

9 Alternating sign matrices J. Striker (NDSU) Resonance February 19, / 69

10 Alternating sign matrices fully-packed loops J. Striker (NDSU) Resonance February 19, / 69

11 Fully-packed loops J. Striker (NDSU) Resonance February 19, / 69

12 Gyration on fully-packed loops The local move on a square J. Striker (NDSU) Resonance February 19, / 69

13 Gyration on fully-packed loops J. Striker (NDSU) Resonance February 19, / 69

14 Gyration on fully-packed loops Apply the local move to all even squares. J. Striker (NDSU) Resonance February 19, / 69

15 Gyration on fully-packed loops Apply the local move to all even squares. J. Striker (NDSU) Resonance February 19, / 69

16 Gyration on fully-packed loops Apply the local move to all even squares. J. Striker (NDSU) Resonance February 19, / 69

17 Gyration on fully-packed loops J. Striker (NDSU) Resonance February 19, / 69

18 Gyration on fully-packed loops Apply the local move to all odd squares. J. Striker (NDSU) Resonance February 19, / 69

19 Gyration on fully-packed loops Apply the local move to all odd squares. J. Striker (NDSU) Resonance February 19, / 69

20 Gyration on fully-packed loops Apply the local move to all odd squares. J. Striker (NDSU) Resonance February 19, / 69

21 Gyration on fully-packed loops J. Striker (NDSU) Resonance February 19, / 69

22 Gyration on fully-packed loops J. Striker (NDSU) Resonance February 19, / 69

23 Resonance of gyration The following theorem of Wieland gives a remarkable property of gyration. Theorem (B. Wieland 2000) Gyration of an FPL rotates the link pattern by an angle of 2π/2n. We reformulate this theorem into a statement of resonance. Corollary Let f be the map from an ASM thru its FPL to the link pattern. Then, (ASM n, < Gyr >, f ) exhibits resonance with frequency 2n. J. Striker (NDSU) Resonance February 19, / 69

24 Resonance of gyration Corollary Let f be the map from an ASM thru its FPL to the link pattern. Then, (ASM n, < Gyr >, f ) exhibits resonance with frequency 2n. ASM n f LP gyr rot ASM n f LP J. Striker (NDSU) Resonance February 19, / 69

25 A length 4 gyration orbit of 5 5 ASMs J. Striker (NDSU) Resonance February 19, / 69

26 A 6 6 ASM with gyration orbit of length J. Striker (NDSU) Resonance February 19, / 69

27 Resonance in orbits of plane partitions and increasing tableaux 1 Resonance defined 2 Resonance of rowmotion 3 Multidimensional promotion and rowmotion 4 Resonance of K-promotion 5 A bijection between increasing tableaux and plane partitions J. Striker (NDSU) Resonance February 19, / 69

28 Posets A poset is a partially ordered set. Definition A poset is a set with a partial order that is reflexive, antisymmetric, and transitive. J. Striker (NDSU) Resonance February 19, / 69

29 Order ideals Definition An order ideal of a poset P is a subset I P such that if y I and z y, then z I. The set of order ideals of P is denoted J(P). J. Striker (NDSU) Resonance February 19, / 69

30 Rowmotion Definition Let P be a poset, and let I J(P). Then rowmotion, Row(I ), is the order ideal generated by the minimal elements of P not in I. An order ideal I J. Striker (NDSU) Resonance February 19, / 69

31 Rowmotion Definition Let P be a poset, and let I J(P). Then rowmotion, Row(I ), is the order ideal generated by the minimal elements of P not in I. Find the minimal elements of P not in I J. Striker (NDSU) Resonance February 19, / 69

32 Rowmotion Definition Let P be a poset, and let I J(P). Then rowmotion, Row(I ), is the order ideal generated by the minimal elements of P not in I. Use them to generate a new order ideal Row(I) J. Striker (NDSU) Resonance February 19, / 69

33 The order of rowmotion in some nice families Theorem (A. Brouwer and A. Schrijver 1974) The order of rowmotion on J(a b) is a + b. Theorem (P. Cameron and D. Fon-der-Flaass 1995) The order of rowmotion on J(a b 2) is a + b + 1. Theorem (P. Cameron and D. Fon-der-Flaass 1995) If a + b + c 1 is prime and c > ab a b + 1, then the cardinality of every orbit of rowmotion on J(a b c) is a multiple of a + b + c 1. J. Striker (NDSU) Resonance February 19, / 69

34 The toggle group Definition For each element e P define its toggle t e : J(P) J(P) as follows. I {e} if e / I and I {e} J(P) t e (I ) = I \ {e} if e I and I \ {e} J(P) I otherwise Definition (P. Cameron and D. Fon-der-Flaass 1995) The toggle group T (J(P)) is the subgroup of the symmetric group S J(P) generated by {t e } e P. J. Striker (NDSU) Resonance February 19, / 69

35 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

36 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

37 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

38 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

39 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

40 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

41 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

42 Rowmotion as a product of toggles Theorem (P. Cameron and D. Fon-der-Flaass 1995) Given any poset P, Row is the toggle group element that toggles the elements in the reverse order of any linear extension. J. Striker (NDSU) Resonance February 19, / 69

43 Promotion and rowmotion Theorem (N. Williams and S. 2012) In any ranked poset, there is an equivariant bijection between the order ideals under rowmotion (toggle top to bottom) and promotion (toggle left to right). Promotion and rowmotion have the same orbit structure! J. Striker (NDSU) Resonance February 19, / 69

44 Partition promotion and rowmotion Theorem (A. Brouwer and A. Schrijver 1974) The order of rowmotion on J(a b) is a + b. Explanation (as a corollary of the theorem on the previous slide): Corollary (N. Williams and S. 2012) There is an equivariant bijection between the order ideals of [a] [b] under rowmotion and binary words of length a + b with n ones under rotation. {1100, 1001, 0011, 0110} {1010, 0101} J. Striker (NDSU) Resonance February 19, / 69

45 the resulting words are balanced. Figure 12 translates the boundary path matrices of Figure 11 to balanced words. We show that this bijection is equivariant, using the definition of ψ in Theorem 7.5. The first rule, ψ [ A] = A, 3,9 corresponds to the case when the first column of the boundary path matrix is (0, 0). This column 2,8 can swap with all other columns without violating the boundary path matrix condition, and so it is 3,7 moved to the end of the word under promotion. 5,7 Consider when 0,6 the first column is (1, 0) 2,6. This column can swap with (0, 0) and (1, 0) without violating the boundary path matrix condition, but it cannot swap with (0, 1) or (1, 1) The second rule, ψ [(A 1,5 1)A 2] = A 1(A 3,5 2), corresponds to when the first column is (1, 0) and the first column it encounters 0,4 that it cannot 2,4 swap with is (0, 1). In this case, the (1, 0) remains fixed, Plane partition promotion and rowmotion 4,8 Theorem (P. Cameron 1,7 and D. Fon-der-Flaass 1995) 4,6 The order of 1,5 rowmotion on J(a b 2) is a + b + 1. Explanation and the (0,(as 1) isafreecorollary to move to theof end of thetheorem word. ] on a previous slide): [(A 1 )( A 2 )(... )( A k )A k+1 The third rule, ψ 1,3 Theorem (N. Williams and S. 2012) = A 1(A 2 )(... )( A k )( A k+1 ), corresponds to when the first column (1, 0) encounters (1, 1) first. Then the (1, 0) remains and the (1, 1) can swap to the right without violating the boundary path matrix condition until it reaches the first (0, 1) such Figure that 9. the On columns the left to is the [2] left [3] have [4] drawn the same as an number rc poset of 1s of height in the 2. top When and bottom there rows. This (0, There is anare 1) equivariant two then elements continues with to the bijection the end same of position, the word. between the second element J(a is raised; b the 2) position under We now is indicated give an equivariant by a dotted bijection arrow down. from Covering β m,n under relations ψ to noncrossing are drawn partitions with solid of black [n + m + 1] rowmotion into m lines, + and 1 blocks andnoncrossing are under projected rotation. down For partitions as solid i < j, gray if ( lines. in of position On [a the + right ib is + paired are1] the with into order ) ideal in b position + 1 blocks j including and boundary brackets from pathsthe corresponding symbol )( then to irightmost and j areplane in a block partition together. in Figure The10resulting under rotation. noncrossing (covering partition relations will between have exactly layers m are + 1suppressed). blocks because there are m s in the bottom row of the boundary path matrix: each (0, 0) column is replaced by a, which corresponds to a singleton block, and each (1, 0) column becomes a (, which corresponds to the first element in a block. For an example, see Figure 12. It is clear that this bijection is equivariant.. { } ( )( ( )( )), ( )( ( )( )), ( ( )( ) )( ), ( )( )( )( ), ( )( )( )( ), ( )( ( )( ) ), (( )( ) )( ), ( )( ) ( )( ). Figure 10. An orbit of J([2] [3] [4]) under promotion { ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} J. Striker Figure (NDSU) 12. The balanced words coming Resonance from the boundary path matrices February in Fig-19, / 69

46 Plane partition resonance What can we say about the orbit structure of rowmotion on J(a b c) for a, b, c > 2? Theorem (K. Dilks, O. Pechenik, S ) (J(a b c), Row, X max D) exhibits resonance with frequency a + b + c 1. In [N. Williams S. 2012], we characterized J(a b c) in terms of boundary path matrices, b (a + b + c 1) matrices with entries in {0, 1} such that the rows each sum to a and satisfy a certain inequality. We noted that Pro traces from left to right through the columns of the matrix, swapping each pair of entries in adjacent columns and the same row that result in a matrix still satisfying the inequality. J. Striker (NDSU) Resonance February 19, / 69

47 Plane partition resonance Given I J(a b c), define X max (I ) to be the vector of length a + b + c 1 whose jth component is the maximum entry in column j of the boundary path matrix. Lemma Let I J(a b c). If X max (I ) = (x 1, x 2,..., x a+b+c 1 ), then X max (Pro(I )) is the cyclic shift (x 2,..., x a+b+c 1, x 1 ). Proposition (K. Dilks, O. Pechenik, S ) (J(a b c), Pro, X max ) exhibits resonance with frequency a + b + c 1. Let D be the toggle group element conjugating Row to Pro. Theorem (K. Dilks, O. Pechenik, S ) (J(a b c), Row, X max D) exhibits resonance with frequency a + b + c 1. J. Striker (NDSU) Resonance February 19, / 69

48 Corollary Suppose a + b + c 1 is prime and I J(a b c). Suppose there is a zero in X max (I ). Then the size of the promotion orbit of I is a multiple of a + b + c 1. Theorem (P. Cameron and D. Fon-der-Flaass 1995) If a + b + c 1 is prime and c > ab a b + 1, then the cardinality of every orbit of rowmotion on J(a b c) is a multiple of a + b + c 1. New proof: If a + b + c 1 is prime and c > ab a b + 1, then there are a total of ab ones in the boundary path matrix, but a total of a + b + c 1 > ab columns in the matrix, so there must be a column of all zeros. Thus, there is a zero in X max (I ) for any I J(a b c), and the promotion orbit is a multiple of a + b + c 1. So the orbits of rowmotion are also multiples of a + b + c 1. J. Striker (NDSU) Resonance February 19, / 69

49 Resonance in orbits of plane partitions and increasing tableaux 1 Resonance defined 2 Resonance of rowmotion 3 Multidimensional promotion and rowmotion 4 Resonance of K-promotion 5 A bijection between increasing tableaux and plane partitions J. Striker (NDSU) Resonance February 19, / 69

50 Multidimensional promotion and rowmotion Definition A lattice projection of a poset P is an order and rank preserving map π : P Z n, where x y in Z n if and only if the component-wise difference y x is in N n. h e f g b c d a J. Striker (NDSU) Resonance February 19, / 69

51 Multidimensional promotion and rowmotion Definition A lattice projection of a poset P is an order and rank preserving map π : P Z n, where x y in Z n if and only if the component-wise difference y x is in N n. h e f g b c d a e b x c a y h f g d z J. Striker (NDSU) Resonance February 19, / 69

52 Multidimensional promotion and rowmotion Definition A lattice projection of a poset P is an order and rank preserving map π : P Z n, where x y in Z n if and only if the component-wise difference y x is in N n. h e f g e h h b c d bc fg bcd efg a a d a J. Striker (NDSU) Resonance February 19, / 69

53 Multidimensional promotion and rowmotion Definition Let P be a poset with an n-dimensional lattice projection π, and let v = (v 1, v 2, v 3,..., v n ), where v j {±1}. Let T i π,v be the product of toggles t x for all elements x of P that lie on the affine hyperplane π(x), v = i. Then define promotion with respect to π and v as Pro π,v =... T 2 π,vt 1 π,vt 0 π,vt 1 π,vt 2 π,v.... Two elements of the poset that lie on the same affine hyperplane π(x), v = i cannot be part of a covering relation, so the operator T i π,v is well-defined and (T i π,v) 2 = 1. J. Striker (NDSU) Resonance February 19, / 69

54 Pro id,(1,1, 1) = T 2 π,vt 1 π,vt 0 π,vt 1 π,vt 2 π,vt 3 π,v J. Striker (NDSU) Resonance February 19, / 69

55 Multidimensional promotion and rowmotion Definition Let P be a poset with an n-dimensional lattice projection π, and let v = (v 1, v 2, v 3,..., v n ), where v j {±1}. Let T i π,v be the product of toggles t x for all elements x of P that lie on the affine hyperplane π(x), v = i. Then define promotion with respect to π and v as Pro π,v =... T 2 π,vt 1 π,vt 0 π,vt 1 π,vt 2 π,v.... Proposition For any finite ranked poset P and lattice projection π, Pro π,(1,1,...,1) = Row. J. Striker (NDSU) Resonance February 19, / 69

56 Multidimensional promotion and rowmotion Theorem (K. Dilks, O. Pechenik, S ) Let P be a finite poset with an n-dimensional lattice projection π. Let v = (v 1, v 2, v 3,..., v n ) and w = (w 1, w 2, w 3,..., w n ), where v j, w j {±1}. Then there is an equivariant bijection between J(P) under Pro π,v and J(P) under Pro π,w. Rowmotion and 2 n 1 other promotions have the same orbit structure! J. Striker (NDSU) Resonance February 19, / 69

57 Multidimensional promotion and rowmotion Theorem (K. Dilks, O. Pechenik, S ) Let P be a finite poset with an n-dimensional lattice projection π. Let v = (v 1, v 2, v 3,..., v n ) and w = (w 1, w 2, w 3,..., w n ), where v j, w j {±1}. Then there is an equivariant bijection between J(P) under Pro π,v and J(P) under Pro π,w. Rowmotion and 2 n 1 other promotions have the same orbit structure! Proof. Similar argument as in [N. Williams S. 2012]. J. Striker (NDSU) Resonance February 19, / 69

58 Resonance in orbits of plane partitions and increasing tableaux 1 Resonance defined 2 Resonance of rowmotion 3 Multidimensional promotion and rowmotion 4 Resonance of K-promotion 5 A bijection between increasing tableaux and plane partitions J. Striker (NDSU) Resonance February 19, / 69

59 Increasing tableaux Definition An increasing tableau of shape λ is a filling of a partition shape λ with positive integers so that labels strictly increase from left to right across rows and from top to bottom down columns. Let Inc q (λ) denote the set of increasing tableaux of shape λ with entries at most q. An increasing tableau in Inc 10 (4, 4, 4, 2): J. Striker (NDSU) Resonance February 19, / 69

60 K-Promotion on an increasing tableau Delete 1 s Fill and decrement J. Striker (NDSU) Resonance February 19, / 69

61 Resonance of K-promotion Define the content of an increasing tableau T Inc q (λ) to be the binary sequence Con(T ) = (a 1, a 2,..., a q ), where a i = 1 if i is an entry of T and a i = 0 if it is not. Lemma Let T Inc q (λ). If Con(T ) = (a 1, a 2,..., a q ), then Con(K-Pro(T )) is the cyclic shift (a 2,..., a q, a 1 ). Theorem (K. Dilks, O. Pechenik, S ) (Inc q (λ), K-Pro, Con) exhibits resonance with frequency q. Corollary Suppose q is prime and T Inc q (λ) does not have full content. Then the size of the K-promotion orbit of T is a multiple of q. J. Striker (NDSU) Resonance February 19, / 69

62 K-Promotion as a product of involutions Proposition For T Inc q (λ), K-Pro(T ) = K-BK q 1 K-BK 1 (T ) K-BK 3 K-BK J. Striker (NDSU) Resonance February 19, / 69

63 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

64 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

65 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

66 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

67 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

68 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

69 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

70 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

71 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

72 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

73 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

74 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

75 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

76 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

77 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

78 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

79 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

80 K-Promotion as a product of involutions J. Striker (NDSU) Resonance February 19, / 69

81 Resonance in orbits of plane partitions and increasing tableaux 1 Resonance defined 2 Resonance of rowmotion 3 Multidimensional promotion and rowmotion 4 Resonance of K-promotion 5 A bijection between increasing tableaux and plane partitions J. Striker (NDSU) Resonance February 19, / 69

82 A bijection between increasing tableaux and plane partitions P = Project to bottom face Rotate Add 1+rank = Ψ 3 (P) J. Striker (NDSU) Resonance February 19, / 69

83 A bijection between increasing tableaux and plane partitions Theorem (K. Dilks, O. Pechenik, S ) Ψ 3 : J(a b c) Inc a+b+c 1 (a b) gives a bijection between plane partitions inside an a b c box and increasing tableaux of shape a b and entries at most a + b + c 1. Proof. The map is defined as the composition of a projection, a rotation, and entrywise addition, all of which are clearly invertible. J. Striker (NDSU) Resonance February 19, / 69

84 A bijection between increasing tableaux and plane partitions P = Project to bottom face Rotate Add 1+rank = Ψ 3 (P) J. Striker (NDSU) Resonance February 19, / 69

85 An equivariant bijection Lemma Ψ 3 intertwines Pro id,(1,1, 1) and K-Pro. That is, the following diagram commutes: J(a b c) Inc a+b+c 1 (a b) Ψ 3 Pro id,(1,1, 1) K-Pro J(a b c) Inc a+b+c 1 (a b) Ψ 3 Theorem (K. Dilks, O. Pechenik, S ) J(a b c) under Row is in equivariant bijection with Inc a+b+c 1 (a b) under K-Pro. J. Striker (NDSU) Resonance February 19, / 69

86 Each hyperplane-toggle corresponds to a K-BK i involution J. Striker (NDSU) Resonance February 19, / 69

87 Corollaries of the equivariant bijection Theorem (K. Dilks, O. Pechenik, S ) J(a b c) under Row is in equivariant bijection with Inc a+b+c 1 (a b) under K-Pro. Corollary There are K-Pro-equivariant bijections between the sets Inc a+b+c 1 (a b), Inc a+b+c 1 (a c), and Inc a+b+c 1 (b c). J. Striker (NDSU) Resonance February 19, / 69

88 J. Striker (NDSU) Resonance February 19, / 69

89 Corollaries of the equivariant bijection Corollary There are K-Pro-equivariant bijections between the sets Inc a+b+c 1 (a b), Inc a+b+c 1 (a c), and Inc a+b+c 1 (b c). Corollary The order of K-Pro on Inc a+b (a b) is a + b. Proof. Using the tri-fold symmetry, there is a K-Pro-equivariant bijection between the sets Inc a+b (a b) and Inc a+b (1 a). The result is then immediate. J. Striker (NDSU) Resonance February 19, / 69

90 Corollaries of the equivariant bijection Corollary The order of K-Pro on Inc a+b+1 (a b) is a + b + 1. Proof. Using the tri-fold symmetry, there is a K-Pro-equivariant bijection between the sets Inc a+b+1 (a b) and Inc a+b+1 (2 a). The result is then immediate by a result of Pechenik on increasing tableaux or by the result of Cameron and Fon-der-Flaass on J(a b 2). J. Striker (NDSU) Resonance February 19, / 69

91 A conjecture Conjecture The order of Row on J(a b 3) is a + b + 2. This is equivalent to the order of K-promotion being a + b + 2 on either Inc a+b+2 (a b) or Inc a+b+2 (3 a). We have verified conjecture for a 7 and b arbitrary. J. Striker (NDSU) Resonance February 19, / 69

92 An improved bound, via our main bijection Theorem (P. Cameron and D. Fon-der-Flaass 1995) If a + b + c 1 is prime and c > ab a b + 1, then the cardinality of every orbit of rowmotion on J(a b c) is a multiple of a + b + c 1. Theorem (K. Dilks, O. Pechenik, S ) If a + b + c 1 is prime and c > 2 3 ab a b + 4 3, then the cardinality of every orbit of Row on J(a b c) is a multiple of a + b + c 1. Proof. By analyzing the corresponding increasing tableaux under K-Pro. J. Striker (NDSU) Resonance February 19, / 69

93 Open problems Problem Construct a natural map f such that (ASM n, SPro, f ) exhibits resonance with frequency 3n 2. Problem Construct a natural map f such that (TSSCPP n, Row, f ) exhibits resonance with frequency 3n 2. J. Striker (NDSU) Resonance February 19, / 69

94 ASM under SPro TSSCPP under Row n 3n 2 Orbit Size Number of Orbits Orbit Size Number of Orbits k, k > * * J. Striker (NDSU) Resonance February 19, / 69

95 Open problems Problem Construct a natural map f such that (ASM n, SPro, f ) exhibits resonance with frequency 3n 2. Problem Construct a natural map f such that (TSSCPP n, Row, f ) exhibits resonance with frequency 3n 2. Problem Construct a natural map f such that (TSPP n, Row, f ) exhibits resonance with frequency 3n 1. J. Striker (NDSU) Resonance February 19, / 69

96 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of rowmotion 3 Multidimensional promotion and rowmotion 4 Resonance of K-promotion 5 A bijection between increasing tableaux and plane partitions J. Striker (NDSU) Resonance February 19, / 69

97 K. Dilks, O. Pechenik, and J. Striker, Resonance in orbits of plane partitions and increasing tableaux, J. Striker and N. Williams, Promotion and rowmotion, Eur. J. Combin. 33 (2012), no. 8, J. Striker (NDSU) Resonance February 19, / 69