Jessica Striker. November 10, 2015

Transcription

1 Resonance in dynamical algebraic combinatorics Jessica Striker joint work with Kevin Dilks (NDSU) and Oliver Pechenik (UIUC) North Dakota State University November 10, 2015 J. Striker (NDSU) Resonance in DAC November 10, / 65

2 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of gyration 3 Resonance of K-promotion 4 Resonance of rowmotion 5 Resonance conjectures J. Striker (NDSU) Resonance in DAC November 10, / 65

3 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of gyration 3 Resonance of K-promotion 4 Resonance of rowmotion 5 Resonance conjectures J. Striker (NDSU) Resonance in DAC November 10, / 65

4 Resonance in musical instruments J. Striker (NDSU) Resonance in DAC November 10, / 65

5 Resonance in differential equations J. Striker (NDSU) Resonance in DAC November 10, / 65

6 Resonance in dynamical algebraic combinatorics J. Striker (NDSU) Resonance in DAC November 10, / 65

7 Resonance in dynamical algebraic combinatorics What is dynamical algebraic combinatorics? J. Striker (NDSU) Resonance in DAC November 10, / 65

8 Resonance in dynamical algebraic combinatorics J. Striker (NDSU) Resonance in DAC November 10, / 65

9 Resonance in dynamical algebraic combinatorics Definition Dynamical algebraic combinatorics is the study of actions on objects important in algebraic combinatorics. J. Striker (NDSU) Resonance in DAC November 10, / 65

10 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 (standard and increasing) tableaux Rowmotion on order ideals of a poset J. Striker (NDSU) Resonance in DAC November 10, / 65

11 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 on a set Y, and f : X Y. 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 in DAC November 10, / 65

12 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 on a set Y, and f : X Y. 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 An example - gyration on alternating sign matrices J. Striker (NDSU) Resonance in DAC November 10, / 65

13 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of gyration 3 Resonance of K-promotion 4 Resonance of rowmotion 5 Resonance conjectures J. Striker (NDSU) Resonance in DAC November 10, / 65

14 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 in DAC November 10, / 65

15 Alternating sign matrices J. Striker (NDSU) Resonance in DAC November 10, / 65

16 Alternating sign matrices fully-packed loops J. Striker (NDSU) Resonance in DAC November 10, / 65

17 Fully-packed loops J. Striker (NDSU) Resonance in DAC November 10, / 65

18 Fully-packed loops Start with an n n grid. J. Striker (NDSU) Resonance in DAC November 10, / 65

19 Fully-packed loops Add boundary conditions. J. Striker (NDSU) Resonance in DAC November 10, / 65

20 Fully-packed loops Interior vertices adjacent to 2 edges. J. Striker (NDSU) Resonance in DAC November 10, / 65

21 Gyration on fully-packed loops Given a square in the grid, the local move swaps the configurations below and leaves every other edge configuration fixed. J. Striker (NDSU) Resonance in DAC November 10, / 65

22 Gyration on fully-packed loops J. Striker (NDSU) Resonance in DAC November 10, / 65

23 Gyration on fully-packed loops Start with the even squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

24 Gyration on fully-packed loops Apply the local move to all even squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

25 Gyration on fully-packed loops Apply the local move to all even squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

26 Gyration on fully-packed loops Apply the local move to all even squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

27 Gyration on fully-packed loops Now consider the odd squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

28 Gyration on fully-packed loops Apply the local move to all odd squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

29 Gyration on fully-packed loops Apply the local move to all odd squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

30 Gyration on fully-packed loops Apply the local move to all odd squares. J. Striker (NDSU) Resonance in DAC November 10, / 65

31 Gyration on fully-packed loops J. Striker (NDSU) Resonance in DAC November 10, / 65

32 Gyration on fully-packed loops J. Striker (NDSU) Resonance in DAC November 10, / 65

33 Resonance of gyration The following theorem of Wieland gives a remarkable property of gyration. Theorem (B. Wieland 2000) Gyration on an FPL rotates the link pattern by a factor 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 in DAC November 10, / 65

34 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 in DAC November 10, / 65

35 A length 4 gyration orbit of 5 5 ASMs J. Striker (NDSU) Resonance in DAC November 10, / 65

36 A 6 6 ASM with gyration orbit of length J. Striker (NDSU) Resonance in DAC November 10, / 65

37 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of gyration 3 Resonance of K-promotion 4 Resonance of rowmotion 5 Resonance conjectures J. Striker (NDSU) Resonance in DAC November 10, / 65

38 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 in DAC November 10, / 65

39 K-Promotion on an increasing tableau Delete 1 s Fill and decrement J. Striker (NDSU) Resonance in DAC November 10, / 65

40 Order of promotion on SYT Definition A standard Young tableau of shape λ is an increasing tableau filled with positive integers {1, 2,..., λ }. Let SYT (λ) denote the set of standard Young tableaux of shape λ. Theorem (Haiman 1992) Every element of SYT (λ) for λ a rectangle is fixed by promotion to the power λ. Example: There are SYT of shape (7, 7, 7, 7, 7), but promotion is of order 7 5 = 35. J. Striker (NDSU) Resonance in DAC November 10, / 65

41 Resonance of K-promotion Definition 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. Proposition 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 ). Corollary (Inc q (λ), K-Pro, Con) exhibits resonance with frequency q. J. Striker (NDSU) Resonance in DAC November 10, / 65

42 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 in DAC November 10, / 65

43 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

44 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

45 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

46 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

47 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

48 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

49 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

50 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

51 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

52 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

53 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

54 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

55 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

56 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

57 K-Promotion as a product of involutions J. Striker (NDSU) Resonance in DAC November 10, / 65

58 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of gyration 3 Resonance of K-promotion 4 Resonance of rowmotion 5 Resonance conjectures J. Striker (NDSU) Resonance in DAC November 10, / 65

59 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 in DAC November 10, / 65

60 Order ideals Definition An order ideal of a poset P is a subset X P such that if y X and z y, then z X. The set of order ideals of P is denoted J(P). J. Striker (NDSU) Resonance in DAC November 10, / 65

61 Rowmotion Definition Let P be a poset, and let X J(P). Then rowmotion, Row(X ), is the order ideal generated by the minimal elements of P not in X. An order ideal X J. Striker (NDSU) Resonance in DAC November 10, / 65

62 Rowmotion Definition Let P be a poset, and let X J(P). Then rowmotion, Row(X ), is the order ideal generated by the minimal elements of P not in X. Find the minimal elements of P not in X J. Striker (NDSU) Resonance in DAC November 10, / 65

63 Rowmotion Definition Let P be a poset, and let X J(P). Then rowmotion, Row(X ), is the order ideal generated by the minimal elements of P not in X. Use them to generate a new order ideal Row(X) J. Striker (NDSU) Resonance in DAC November 10, / 65

64 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 in DAC November 10, / 65

65 The toggle group Definition For each element e P define its toggle t e : J(P) J(P) as follows. X {e} if e / X and X {e} J(P) t e (X ) = X \ {e} if e X and X \ {e} J(P) X 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 in DAC November 10, / 65

66 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

67 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

68 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

69 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

70 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

71 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

72 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

73 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, that is, from top to bottom. J. Striker (NDSU) Resonance in DAC November 10, / 65

74 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 in DAC November 10, / 65

75 Partition promotion and rowmotion Theorem (A. Brouwer 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 in DAC November 10, / 65

76 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 D. 1,7 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 wordsresonance coming from in DAC the boundary path matrices November in Fig-10, / 65

77 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. Definition (K. Dilks, O. Pechenik, S ) Let P be a poset with an n-dimensional lattice projection π, and choose a distinguished vector v = (v 1, v 2,..., v n ), where v j {±1}. Let T i π,v be the product of toggles t x for all elements of the poset x that lie on the affine hyperplane π(x), v = i. Then define Pro π,v =... T 2 π,vt 1 π,vt 0 π,vt 1 π,vt 2 π,v.... J. Striker (NDSU) Resonance in DAC November 10, / 65

78 Multidimensional promotion and rowmotion J. Striker (NDSU) Resonance in DAC November 10, / 65

79 Multidimensional promotion and rowmotion Definition (K. Dilks, O. Pechenik, S ) Let P be a poset with an n-dimensional lattice projection π, and choose a distinguished vector v = (v 1, v 2,..., v n ), where v j {±1}. Let T i π,v be the product of toggles t x for all elements of the poset x that lie on the affine hyperplane π(x), v = i. Then define Pro π,v =... T 2 π,vt 1 π,vt 0 π,vt 1 π,vt 2 π,v.... Proposition (K. Dilks, O. Pechenik, S ) For any finite ranked poset P and lattice projection π, Pro π,(1,1,...,1) = Row. J. Striker (NDSU) Resonance in DAC November 10, / 65

80 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}. There is an equivariant bijection between J(P) under Pro π,v and J(P) under Pro π,w. J. Striker (NDSU) Resonance in DAC November 10, / 65

81 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. J. Striker (NDSU) Resonance in DAC November 10, / 65

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 in DAC November 10, / 65

83 An equivariant bijection between increasing tableaux and plane partitions Theorem (K. Dilks, O. Pechenik, S ) 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 J. Striker (NDSU) Resonance in DAC November 10, / 65

84 An equivariant bijection between increasing tableaux and plane partitions Theorem (K. Dilks, O. Pechenik, S ) 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 Why? J(a b c) Inc a+b+c 1 (a b) Ψ 3 J. Striker (NDSU) Resonance in DAC November 10, / 65

85 Each hyperplane-toggle corresponds to a K-BK i involution J. Striker (NDSU) Resonance in DAC November 10, / 65

86 Corollaries of the equivariant bijection between increasing tableaux and plane partitions Corollary (K. Dilks, O. Pechenik, S ) J(a b c) under Row is in equivariant bijection with Inc a+b+c 1 (a c) under K-Pro. Corollary (K. Dilks, O. Pechenik, S ) (J(a b c), Row, Con Ψ 3 ) exhibits resonance with frequency a + b + c 1. J. Striker (NDSU) Resonance in DAC November 10, / 65

87 More corollaries of the equivariant bijection between increasing tableaux and plane partitions 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 in DAC November 10, / 65

88 J. Striker (NDSU) Resonance in DAC November 10, / 65

89 Even more corollaries of the equivariant bijection between increasing tableaux and plane partitions 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 in DAC November 10, / 65

90 Even more corollaries of the equivariant bijection between increasing tableaux and plane partitions 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+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. J. Striker (NDSU) Resonance in DAC November 10, / 65

91 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of gyration 3 Resonance of K-promotion 4 Resonance of rowmotion 5 Resonance conjectures J. Striker (NDSU) Resonance in DAC November 10, / 65

92 Resonance conjectures Conjecture There exists a map f such that (ASM n, superpromotion, f ) exhibits resonance with frequency 3n 2. Conjecture There exists a map f such that (TSSCPP n, rowmotion, f ) exhibits resonance with frequency 3n 2. J. Striker (NDSU) Resonance in DAC November 10, / 65

93 Data supporting these resonance conjectures 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 in DAC November 10, / 65

94 Resonance in dynamical algebraic combinatorics 1 Resonance defined 2 Resonance of gyration 3 Resonance of K-promotion 4 Resonance of rowmotion 5 Resonance conjectures J. Striker (NDSU) Resonance in DAC November 10, / 65

95 NDSU Math FORWARD K. Dilks, O. Pechenik, and J. Striker, Resonance in orbits of plane partitions, in preparation. J. Striker and N. Williams, Promotion and rowmotion, Eur. J. Combin. 33 (2012), no. 8, J. Striker (NDSU) Resonance in DAC November 10, / 65