Resonance in orbits of plane partitions and increasing tableaux. Jessica Striker North Dakota State University
|
|
- Austen Fox
- 5 years ago
- Views:
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
Jessica Striker. November 10, 2015
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
More informationResonance in orbits of plane partitions
FPSAC 2016 Vancouver, Canada DMTCS proc. BC, 2016, 383 394 Resonance in orbits of plane partitions Kevin Dilks 1, Oliver Pechenik 2, and Jessica Striker 1 1 Department of Mathematics, North Dakota State
More informationThe long way home. Orbits of plane partitions. Oliver Pechenik. University of Michigan. UM Undergraduate Math Club October 2017
Taking the long way home Orbits of plane partitions University of Michigan UM Undergraduate Math Club October 2017 Mostly based on joint work with Kevin Dilks and Jessica Striker (NDSU) motion of partitions
More informationThe toggle group, homomesy, and the Razumov-Stroganov correspondence
The toggle group, homomesy, and the Razumov-Stroganov correspondence Jessica Striker Department of Mathematics North Dakota State University Fargo, North Dakota, U.S.A. jessica.striker@ndsu.edu Submitted:
More informationPromotion and Rowmotion
FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 273 28 Promotion and Rowmotion Jessica Striker and Nathan Williams School of Mathematics, University of Minnesota, 20 Church St. SE, Minneapolis, MN Abstract.
More informationOrbits of plane partitions of exceptional Lie type
(University of Michigan) Joint Mathematics Meetings, San Diego January 2018 Based on joint work with Holly Mandel (Berkeley) arxiv:1712.09180 Minuscule posets The minuscule posets are the following 5 families:
More informationA (lattice) path formula for birational rowmotion on a product of two chains
A (lattice) path formula for birational rowmotion on a product of two chains Tom Roby (UConn) Describing joint research with Gregg Musiker (University of Minnesota) Workshop on Computer Algebra in Combinatorics
More informationDynamical algebraic combinatorics
Dynamical algebraic combinatorics organized by James Propp, Tom Roby, Jessica Striker, and Nathan Williams Workshop Summary Topics of the Workshop This workshop centered around dynamical systems arising
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationarxiv: v2 [math.co] 12 Dec 2018
WHIRLING INJECTIONS, SURJECTIONS, AND OTHER FUNCTIONS BETWEEN FINITE SETS MICHAEL JOSEPH, JAMES PROPP, AND TOM ROBY arxiv:1711.02411v2 [math.co] 12 Dec 2018 Abstract. This paper analyzes a certain action
More informationMULTI-ORDERED POSETS. Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A06 MULTI-ORDERED POSETS Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States lbishop@oxy.edu
More informationStrange Combinatorial Connections. Tom Trotter
Strange Combinatorial Connections Tom Trotter Georgia Institute of Technology trotter@math.gatech.edu February 13, 2003 Proper Graph Colorings Definition. A proper r- coloring of a graph G is a map φ from
More informationCombinatorial Structures
Combinatorial Structures Contents 1 Permutations 1 Partitions.1 Ferrers diagrams....................................... Skew diagrams........................................ Dominance order......................................
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationarxiv: v1 [math.co] 3 Nov 2014
SPARSE MATRICES DESCRIBING ITERATIONS OF INTEGER-VALUED FUNCTIONS BERND C. KELLNER arxiv:1411.0590v1 [math.co] 3 Nov 014 Abstract. We consider iterations of integer-valued functions φ, which have no fixed
More information(II.B) Basis and dimension
(II.B) Basis and dimension How would you explain that a plane has two dimensions? Well, you can go in two independent directions, and no more. To make this idea precise, we formulate the DEFINITION 1.
More informationPROBLEMS FROM CCCC LIX
PROBLEMS FROM CCCC LIX The following is a list of problems from the 59th installment of the Cambridge Combinatorics and Coffee Club, held at the Worldwide Center of Mathematics in Cambridge, MA on February
More information6 Cosets & Factor Groups
6 Cosets & Factor Groups The course becomes markedly more abstract at this point. Our primary goal is to break apart a group into subsets such that the set of subsets inherits a natural group structure.
More informationTactical Decompositions of Steiner Systems and Orbits of Projective Groups
Journal of Algebraic Combinatorics 12 (2000), 123 130 c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Tactical Decompositions of Steiner Systems and Orbits of Projective Groups KELDON
More informationExamples of Groups
Examples of Groups 8-23-2016 In this section, I ll look at some additional examples of groups. Some of these will be discussed in more detail later on. In many of these examples, I ll assume familiar things
More informationSpectra of Semidirect Products of Cyclic Groups
Spectra of Semidirect Products of Cyclic Groups Nathan Fox 1 University of Minnesota-Twin Cities Abstract The spectrum of a graph is the set of eigenvalues of its adjacency matrix A group, together with
More informationNote that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.
Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions
More informationWeek 4-5: Generating Permutations and Combinations
Week 4-5: Generating Permutations and Combinations February 27, 2017 1 Generating Permutations We have learned that there are n! permutations of {1, 2,...,n}. It is important in many instances to generate
More informationSupplement to Multiresolution analysis on the symmetric group
Supplement to Multiresolution analysis on the symmetric group Risi Kondor and Walter Dempsey Department of Statistics and Department of Computer Science The University of Chicago risiwdempsey@uchicago.edu
More informationPoset and Polytope Perspectives On Alternating Sign Matrices
Poset and Polytope Perspectives On Alternating Sign Matrices A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Jessica Palencia Striker IN PARTIAL FULFILLMENT
More informationSquare 2-designs/1. 1 Definition
Square 2-designs Square 2-designs are variously known as symmetric designs, symmetric BIBDs, and projective designs. The definition does not imply any symmetry of the design, and the term projective designs,
More informationDISTINGUISHING PARTITIONS AND ASYMMETRIC UNIFORM HYPERGRAPHS
DISTINGUISHING PARTITIONS AND ASYMMETRIC UNIFORM HYPERGRAPHS M. N. ELLINGHAM AND JUSTIN Z. SCHROEDER In memory of Mike Albertson. Abstract. A distinguishing partition for an action of a group Γ on a set
More informationGroups of Prime Power Order with Derived Subgroup of Prime Order
Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*
More information5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX
5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX TANTELY A. RAKOTOARISOA 1. Introduction In statistical mechanics, one studies models based on the interconnections between thermodynamic
More informationHigher Spin Alternating Sign Matrices
Higher Spin Alternating Sign Matrices Roger E. Behrend and Vincent A. Knight School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK behrendr@cardiff.ac.uk, knightva@cardiff.ac.uk Submitted: Aug
More informationSection Summary. Relations and Functions Properties of Relations. Combining Relations
Chapter 9 Chapter Summary Relations and Their Properties n-ary Relations and Their Applications (not currently included in overheads) Representing Relations Closures of Relations (not currently included
More informationFully Packed Loops Model: Integrability and Combinatorics. Plan
ully Packed Loops Model 1 Fully Packed Loops Model: Integrability and Combinatorics Moscow 05/04 P. Di Francesco, P. Zinn-Justin, Jean-Bernard Zuber, math.co/0311220 J. Jacobsen, P. Zinn-Justin, math-ph/0402008
More informationMath 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9
Math 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9 Section 0. Sets and Relations Subset of a set, B A, B A (Definition 0.1). Cartesian product of sets A B ( Defintion 0.4). Relation (Defintion 0.7). Function,
More informationNoncrossing partitions, toggles, and homomesies
Noncrossing partitions, toggles, and homomesies The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Einstein,
More informationAUTOMORPHISM GROUPS AND SPECTRA OF CIRCULANT GRAPHS
AUTOMORPHISM GROUPS AND SPECTRA OF CIRCULANT GRAPHS MAX GOLDBERG Abstract. We explore ways to concisely describe circulant graphs, highly symmetric graphs with properties that are easier to generalize
More informationMatrix Algebra 2.1 MATRIX OPERATIONS Pearson Education, Inc.
2 Matrix Algebra 2.1 MATRIX OPERATIONS MATRIX OPERATIONS m n If A is an matrixthat is, a matrix with m rows and n columnsthen the scalar entry in the ith row and jth column of A is denoted by a ij and
More informationCoxeter-Knuth Classes and a Signed Little Bijection
Coxeter-Knuth Classes and a Signed Little Bijection Sara Billey University of Washington Based on joint work with: Zachary Hamaker, Austin Roberts, and Benjamin Young. UC Berkeley, February, 04 Outline
More information5 Group theory. 5.1 Binary operations
5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1
More informationDefinition: A binary relation R from a set A to a set B is a subset R A B. Example:
Chapter 9 1 Binary Relations Definition: A binary relation R from a set A to a set B is a subset R A B. Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B.
More informationA DECOMPOSITION OF SCHUR FUNCTIONS AND AN ANALOGUE OF THE ROBINSON-SCHENSTED-KNUTH ALGORITHM
A DECOMPOSITION OF SCHUR FUNCTIONS AND AN ANALOGUE OF THE ROBINSON-SCHENSTED-KNUTH ALGORITHM S. MASON Abstract. We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline
More informationIsomorphisms between pattern classes
Journal of Combinatorics olume 0, Number 0, 1 8, 0000 Isomorphisms between pattern classes M. H. Albert, M. D. Atkinson and Anders Claesson Isomorphisms φ : A B between pattern classes are considered.
More informationWieland drift for triangular fully packed loop configurations
Wieland drift for triangular fully packed loop configurations Sabine Beil Ilse Fischer Fakultät für Mathematik Universität Wien Wien, Austria {sabine.beil,ilse.fischer}@univie.ac.at Philippe Nadeau Institut
More informationINTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS
INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS HANMING ZHANG Abstract. In this paper, we will first build up a background for representation theory. We will then discuss some interesting topics in
More informationPermutation Groups and Transformation Semigroups Lecture 2: Semigroups
Permutation Groups and Transformation Semigroups Lecture 2: Semigroups Peter J. Cameron Permutation Groups summer school, Marienheide 18 22 September 2017 I am assuming that you know what a group is, but
More informationFoundations Revision Notes
oundations Revision Notes hese notes are designed as an aid not a substitute for revision. A lot of proofs have not been included because you should have them in your notes, should you need them. Also,
More informationcan only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4
.. Conditions for Injectivity and Surjectivity In this section, we discuss what we can say about linear maps T : R n R m given only m and n. We motivate this problem by looking at maps f : {,..., n} {,...,
More informationWhat is a semigroup? What is a group? What is the difference between a semigroup and a group?
The second exam will be on Thursday, July 5, 2012. The syllabus will be Sections IV.5 (RSA Encryption), III.1, III.2, III.3, III.4 and III.8, III.9, plus the handout on Burnside coloring arguments. Of
More informationGROUP THEORY PRIMER. New terms: tensor, rank k tensor, Young tableau, Young diagram, hook, hook length, factors over hooks rule
GROUP THEORY PRIMER New terms: tensor, rank k tensor, Young tableau, Young diagram, hook, hook length, factors over hooks rule 1. Tensor methods for su(n) To study some aspects of representations of a
More informationALGEBRAIC GEOMETRY I - FINAL PROJECT
ALGEBRAIC GEOMETRY I - FINAL PROJECT ADAM KAYE Abstract This paper begins with a description of the Schubert varieties of a Grassmannian variety Gr(k, n) over C Following the technique of Ryan [3] for
More informationThe Littlewood-Richardson Rule
REPRESENTATIONS OF THE SYMMETRIC GROUP The Littlewood-Richardson Rule Aman Barot B.Sc.(Hons.) Mathematics and Computer Science, III Year April 20, 2014 Abstract We motivate and prove the Littlewood-Richardson
More information11 Block Designs. Linear Spaces. Designs. By convention, we shall
11 Block Designs Linear Spaces In this section we consider incidence structures I = (V, B, ). always let v = V and b = B. By convention, we shall Linear Space: We say that an incidence structure (V, B,
More informationCentral Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J
Central Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J Frank Curtis, John Drew, Chi-Kwong Li, and Daniel Pragel September 25, 2003 Abstract We study central groupoids, central
More informationAlgebraic Methods in Combinatorics
Algebraic Methods in Combinatorics Po-Shen Loh 27 June 2008 1 Warm-up 1. (A result of Bourbaki on finite geometries, from Răzvan) Let X be a finite set, and let F be a family of distinct proper subsets
More informationInvolutions by Descents/Ascents and Symmetric Integral Matrices. Alan Hoffman Fest - my hero Rutgers University September 2014
by Descents/Ascents and Symmetric Integral Matrices Richard A. Brualdi University of Wisconsin-Madison Joint work with Shi-Mei Ma: European J. Combins. (to appear) Alan Hoffman Fest - my hero Rutgers University
More informationJónsson posets and unary Jónsson algebras
Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal
More informationSYMMETRIES IN R 3 NAMITA GUPTA
SYMMETRIES IN R 3 NAMITA GUPTA Abstract. This paper will introduce the concept of symmetries being represented as permutations and will proceed to explain the group structure of such symmetries under composition.
More informationCylindric Young Tableaux and their Properties
Cylindric Young Tableaux and their Properties Eric Neyman (Montgomery Blair High School) Mentor: Darij Grinberg (MIT) Fourth Annual MIT PRIMES Conference May 17, 2014 1 / 17 Introduction Young tableaux
More informationMultiplicity Free Expansions of Schur P-Functions
Annals of Combinatorics 11 (2007) 69-77 0218-0006/07/010069-9 DOI 10.1007/s00026-007-0306-1 c Birkhäuser Verlag, Basel, 2007 Annals of Combinatorics Multiplicity Free Expansions of Schur P-Functions Kristin
More informationAn Investigation on an Extension of Mullineux Involution
An Investigation on an Extension of Mullineux Involution SPUR Final Paper, Summer 06 Arkadiy Frasinich Mentored by Augustus Lonergan Project Suggested By Roman Bezrukavnikov August 3, 06 Abstract In this
More information7 Matrix Operations. 7.0 Matrix Multiplication + 3 = 3 = 4
7 Matrix Operations Copyright 017, Gregory G. Smith 9 October 017 The product of two matrices is a sophisticated operations with a wide range of applications. In this chapter, we defined this binary operation,
More informationDiscrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009
Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009 Our main goal is here is to do counting using functions. For that, we
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.
MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection
More information* 8 Groups, with Appendix containing Rings and Fields.
* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that
More informationParameterizing orbits in flag varieties
Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.
More informationMore about partitions
Partitions 2.4, 3.4, 4.4 02 More about partitions 3 + +, + 3 +, and + + 3 are all the same partition, so we will write the numbers in non-increasing order. We use greek letters to denote partitions, often
More informationDefinition 2.3. We define addition and multiplication of matrices as follows.
14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationarxiv: v1 [math.co] 6 Jun 2014
WIELAND GYRATION FOR TRIANGULAR FULLY PACKED LOOP CONFIGURATIONS SABINE BEIL, ILSE FISCHER, AND PHILIPPE NADEAU arxiv:1406.1657v1 [math.co] 6 Jun 2014 Abstract. Triangular fully packed loop configurations
More informationEquality of P-partition Generating Functions
Bucknell University Bucknell Digital Commons Honors Theses Student Theses 2011 Equality of P-partition Generating Functions Ryan Ward Bucknell University Follow this and additional works at: https://digitalcommons.bucknell.edu/honors_theses
More informationarxiv: v1 [math.co] 2 Dec 2008
An algorithmic Littlewood-Richardson rule arxiv:08.0435v [math.co] Dec 008 Ricky Ini Liu Massachusetts Institute of Technology Cambridge, Massachusetts riliu@math.mit.edu June, 03 Abstract We introduce
More informationCombinatorics of non-associative binary operations
Combinatorics of non-associative binary operations Jia Huang University of Nebraska at Kearney E-mail address: huangj2@unk.edu December 26, 2017 This is joint work with Nickolas Hein (Benedictine College),
More informationELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS
ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need
More informationBirational Rowmotion: order, homomesy, and cluster connections
Birational Rowmotion: order, homomesy, and cluster connections Tom Roby (University of Connecticut) Describing joint research with Darij Grinberg Combinatorics Seminar University of Minnesota Minneapolis,
More informationActions and Identities on Set Partitions
Actions and Identities on Set Partitions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Marberg,
More informationLinear Algebra Homework and Study Guide
Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More informationQUIVERS AND LATTICES.
QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given
More informationSUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS
SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a partially ordered set P, let Co(P) denote the lattice of all order-convex
More informationLOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi
LOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi Department of Mathematics, East China Normal University, Shanghai, 200062, China and Center for Combinatorics,
More informationDISCRETE SUBGROUPS, LATTICES, AND UNITS.
DISCRETE SUBGROUPS, LATTICES, AND UNITS. IAN KIMING 1. Discrete subgroups of real vector spaces and lattices. Definitions: A lattice in a real vector space V of dimension d is a subgroup of form: Zv 1
More informationPresentation of the Motzkin Monoid
Presentation of the Motzkin Monoid Kris Hatch UCSB June 3, 2012 1 Acknowledgements The mathematical content described in this thesis is derived from a collaboration over the summer at the REU program here
More informationChapter One. Affine Coxeter Diagrams
Chapter One Affine Coxeter Diagrams By the results summarized in Chapter VI, Section 43, of [3], affine Coxeter groups can be characterized as groups generated by reflections of an affine space (by which
More informationDefinitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch
Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary
More informationHomomesy of Alignments in Perfect Matchings
Homomesy of Alignments in Perfect Matchings Ingrid Zhang under the direction of Sam Hopkins Department of Mathematics Massachusetts Institute of Technology Research Science Institute July 0, 2014 Abstract
More informationThe Witt designs, Golay codes and Mathieu groups
The Witt designs, Golay codes and Mathieu groups 1 The Golay codes Let V be a vector space over F q with fixed basis e 1,..., e n. A code C is a subset of V. A linear code is a subspace of V. The vector
More informationExtended 1-perfect additive codes
Extended 1-perfect additive codes J.Borges, K.T.Phelps, J.Rifà 7/05/2002 Abstract A binary extended 1-perfect code of length n + 1 = 2 t is additive if it is a subgroup of Z α 2 Zβ 4. The punctured code
More informationUnless otherwise specified, V denotes an arbitrary finite-dimensional vector space.
MAT 90 // 0 points Exam Solutions Unless otherwise specified, V denotes an arbitrary finite-dimensional vector space..(0) Prove: a central arrangement A in V is essential if and only if the dual projective
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationGENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS. 0. Introduction
Acta Math. Univ. Comenianae Vol. LXV, 2(1996), pp. 247 279 247 GENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS J. HEDLÍKOVÁ and S. PULMANNOVÁ Abstract. A difference on a poset (P, ) is a partial binary
More informationENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.
ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.
More informationReal Analysis Prelim Questions Day 1 August 27, 2013
Real Analysis Prelim Questions Day 1 August 27, 2013 are 5 questions. TIME LIMIT: 3 hours Instructions: Measure and measurable refer to Lebesgue measure µ n on R n, and M(R n ) is the collection of measurable
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationSynthetic Geometry. 1.4 Quotient Geometries
Synthetic Geometry 1.4 Quotient Geometries Quotient Geometries Def: Let Q be a point of P. The rank 2 geometry P/Q whose "points" are the lines of P through Q and whose "lines" are the hyperplanes of of
More informationFINITE GROUP THEORY: SOLUTIONS FALL MORNING 5. Stab G (l) =.
FINITE GROUP THEORY: SOLUTIONS TONY FENG These are hints/solutions/commentary on the problems. They are not a model for what to actually write on the quals. 1. 2010 FALL MORNING 5 (i) Note that G acts
More informationRings and Fields Theorems
Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)
More informationCrossings and Nestings in Tangled Diagrams
Crossings and Nestings in Tangled Diagrams William Y. C. Chen 1, Jing Qin 2 and Christian M. Reidys 3 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071, P. R. China 1 chen@nankai.edu.cn,
More informationDefinition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson
Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies.
More informationA prolific construction of strongly regular graphs with the n-e.c. property
A prolific construction of strongly regular graphs with the n-e.c. property Peter J. Cameron and Dudley Stark School of Mathematical Sciences Queen Mary, University of London London E1 4NS, U.K. Abstract
More informationPermutation groups/1. 1 Automorphism groups, permutation groups, abstract
Permutation groups Whatever you have to do with a structure-endowed entity Σ try to determine its group of automorphisms... You can expect to gain a deep insight into the constitution of Σ in this way.
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More information