discretization of new weyl group orbit functions
play

Discretization of new Weyl group orbit functions J. Hrivn ak, L. - PowerPoint PPT Presentation

Dec 19, 2023 •135 likes •1.16k views

Discretization of new Weyl group orbit functions J. Hrivn ak, L. Motlochov a, J. Patera Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, B rehov a 7, 115 19 Praha 1 Centre de recherches

  1. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − and S l − functions for σ ∈ { 1 , σ e , σ s , σ l } , b ∈ P are the complex functions b : R n → C ϕ σ � ϕ σ σ ( w ) e 2 πi � wb, a � , a ∈ R n b ( a ) = w ∈ W σ = σ e . . . S − functions (known from the Weyl character formula) σ = 1 . . . C − functions σ = σ s . . . S s − functions σ = σ l . . . S l − functions (anti)symmetry with respect to w ∈ W ϕ s b ( wa ) = σ s ( w ) ϕ s b ( a ) ϕ s wb ( a ) = σ s ( w ) ϕ s b ( a ) invariance with respect to shifts from q ∨ ∈ Q ∨ ϕ s b ( a + q ∨ ) = ϕ s b ( a ) J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  2. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  3. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  4. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  5. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  6. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  7. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  8. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  9. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  10. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  11. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  12. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  13. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  14. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  15. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  16. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  17. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  18. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  19. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  20. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  21. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  22. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  23. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  24. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  25. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  26. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  27. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  28. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  29. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  30. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  31. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  32. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  33. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  34. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  35. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  36. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  37. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  38. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  39. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  40. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  41. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  42. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  43. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  44. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  45. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  46. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  47. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  48. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  49. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  50. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  51. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  52. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  53. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  54. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  55. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (1 , 0) ( x, y ) and ϕ s (1 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  56. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (2 , 0) ( x, y ) and ϕ s (2 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  57. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (3 , 1) ( x, y ) and ϕ s (3 , 2) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  58. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (0 , 1) ( x, y ) and ϕ l (1 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  59. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (1 , 2) ( x, y ) and ϕ l (2 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  60. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (2 , 3) ( x, y ) and ϕ l (3 , 2) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  61. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (0 , 1) ( x, y ) and ϕ s (1 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  62. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (1 , 2) ( x, y ) and ϕ s (2 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  63. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (2 , 3) ( x, y ) and ϕ s (3 , 2) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  64. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (1 , 0) ( x, y ) and ϕ l (1 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  65. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (1 , 2) ( x, y ) and ϕ l (2 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  66. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (2 , 3) ( x, y ) and ϕ l (3 , 2) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  67. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Outline 1 Short and long orbit functions Lie groups/Lie algebras Sign homomorphisms S s − and S l − functions 2 Discretization of orbit functions Grids F s M and F l M Grids Λ s M and Λ l M Discrete orthogonality of S s − and S l − functions J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  68. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M M P ∨ 1 W -invariant lattice M P ∨ /Q ∨ 1 M ∈ N , W -invariant finite group M P ∨ /Q ∨ is cM n 1 number of elements of The grid F M F M ≡ 1 M P ∨ /Q ∨ ∩ F The grids F s M ⊂ F M and F l M ⊂ F M M ≡ 1 M P ∨ /Q ∨ ∩ F s F s M ≡ 1 M P ∨ /Q ∨ ∩ F l F l the symbols u s i , u l i ∈ R , i = 0 , . . . , n : u s u l i ∈ Z ≥ 0 , r i ∈ R s i ∈ N , u s i ∈ Z ≥ 0 , u l r i ∈ R l i ∈ N , J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  69. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s 4 and F l 4 of C 2 η α 2 = α ∨ ω 2 = ω ∨ ξ = ω ∨ 2 2 1 H l H s F ω 1 = 1 2 ω ∨ 1 α ∨ α 1 1 r 0 r 2 r 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  70. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s 4 and F l 4 of C 2 η α 2 = α ∨ ω 2 = ω ∨ ξ = ω ∨ 2 2 1 H l H s F ω 1 = 1 2 ω ∨ 1 α ∨ α 1 1 r 0 r 2 r 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  71. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M the explicit form of F s and F l : � � u s � � 1 + · · · + u s � F s M ω ∨ 1 M ω ∨ n � u s 0 + u s 1 m 1 + · · · + u s M = n m n = M n � u l � � � 1 + · · · + u l � F l M ω ∨ 1 M ω ∨ n � u l 0 + u l 1 m 1 + · · · + u l M = n m n = M n Proposition Let m s and m l be the short and long Coxeter numbers, respectively. Then   M < m s M < m l  0  0   | F s | F l M | = M = m s , M | = M = m l 1 1     M > m s . M > m l . | F M − m s | | F M − m l | J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  72. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M Theorem The numbers of points of grids F s M and F l M of Lie algebras B n , C n are given by the following relations. C n , n ≥ 2, 1 � k + 1 � � k � | F s 2 k ( C n ) | = + n n � k + 1 � | F s 2 k +1 ( C n ) | = 2 n � n + k − 1 � � n + k − 2 � | F l 2 k ( C n ) | = + n n � n + k − 1 � | F l 2 k +1 ( C n ) | = 2 n B n , n ≥ 3, 2 | F s M ( B n ) | = | F l M ( C n ) | | F l M ( B n ) | = | F s M ( C n ) | J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  73. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M Theorem The numbers of points of grids F s M and F l M of Lie algebra G 2 are given by the following relations. 6 k +1 ( G 2 ) | = 3 k 2 + k | F s 6 k ( G 2 ) | = 3 k 2 , | F s 6 k +2 ( G 2 ) | = 3 k 2 + 2 k, 6 k +3 ( G 2 ) | = 3 k 2 + 3 k + 1 | F s | F s 6 k +4 ( G 2 ) | = 3 k 2 + 4 k + 1 , 6 k +5 ( G 2 ) | = 3 k 2 + 5 k + 2 . | F s | F s | F l M ( G 2 ) | = | F s M ( G 2 ) | F 4 : the counting formula is also known J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  74. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Outline 1 Short and long orbit functions Lie groups/Lie algebras Sign homomorphisms S s − and S l − functions 2 Discretization of orbit functions Grids F s M and F l M Grids Λ s M and Λ l M Discrete orthogonality of S s − and S l − functions J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  75. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions The dual Lie algebra ∆ → ∆ ∨ : 2 α i α ∨ i = � α i , α i � , i ∈ { 1 , . . . , n } the set ∆ ∨ = { α ∨ 1 , . . . , α ∨ n } is a system of simple roots of some simple Lie algebra span R ∆ ∨ = R n the highest dual root η ≡ − α ∨ 0 = m ∨ 1 α ∨ 1 + · · · + m ∨ n α ∨ n m ∨ j . . . the dual marks of G the dual Cartan matrix C ∨ ij = 2 � α ∨ i , α ∨ j � C ∨ j � = C ji , i, j ∈ { 1 , . . . , n } � α ∨ j , α ∨ J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  76. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Dual affine Weyl group the group generated by n reflections r α ∨ = r α , α ∨ ∈ ∆ ∨ coincides with W W aff is generated by reflections R ∨ = { r ∨ � 0 , r 1 , . . . , r n } , where 2 η r η a = a − 2 � a, η � r ∨ a ∈ R n 0 a = r η a + � η, η � , � η, η � η, Dual affine Weyl group � W aff W aff = Q ⋊ W � the fundamental domain F ∨ of � W aff � � 0 , ω 1 , . . . , ω n F ∨ = m ∨ m ∨ 1 n κ a decomposition of R ∨ = R s ∨ ∪ R l ∨ : R s ∨ = { r α | α ∈ ∆ s } ∪ { r ∨ 0 } R l ∨ = { r α | α ∈ ∆ l } J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  77. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Dual fundamental domains two subsets of boundaries of F ∨ : H s ∨ = { a ∈ F ∨ | ( ∃ r ∈ R s ∨ )( ra = a ) } � � H l ∨ = a ∈ F ∨ | ( ∃ r ∈ R l ∨ )( ra = a ) fundamental domains F s ∨ ⊂ F ∨ , F l ∨ ⊂ F ∨ F s ∨ = F ∨ \ H s ∨ F l ∨ = F ∨ \ H l ∨ the symbols z s i , z l i ∈ R , i = 0 , . . . , n z s z l r i ∈ R s ∨ i > 0 , i ≥ 0 , z s z l r i ∈ R l ∨ i ≥ 0 , i > 0 , � � � � F s ∨ = � z s 1 ω 1 + · · · + z s � z s 0 + z s 1 m ∨ 1 + · · · + z s n m ∨ n ω n n = 1 � � � � F l ∨ = � z l 1 ω 1 + · · · + z l � z l 0 + z l 1 m ∨ 1 + · · · + z l n m ∨ n ω n n = 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  78. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids Λ s M and Λ l M M ∈ N , W -invariant lattice P W -invariant finite group P/MQ number of elements of P/MQ is cM n The grid Λ M Λ M ≡ MF ∨ ∩ P/MQ The grids Λ s M ⊂ Λ M and Λ l M ⊂ Λ M M ≡ MF s ∨ ∩ P/MQ Λ s M ≡ MF l ∨ ∩ P/MQ Λ l Proposition For the numbers of elements of the sets Λ s M and Λ l M it holds that | Λ s M | = | F s M | , | Λ l M | = | F l M | . J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  79. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s ∨ and F l ∨ of C 2 4 4 4 α 2 r ∨ η 0 , 4 4 H s ∨ 4 F ∨ 4 H l ∨ α 2 = α ∨ ω 2 = ω ∨ 2 2 ξ = ω ∨ 1 1 r ∨ 2 ω 2 0 F ∨ ω 1 α ∨ α 1 4 α 1 1 r 2 r 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  80. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete orthogonality of S s − and S l − functions functions ϕ s , ϕ l on the grids F s M and F l M ϕ s b + MQ ( u ) = ϕ s b ( u ) , u ∈ F s M ϕ l b + MQ ( u ) = ϕ l b ( u ) , u ∈ F l M ϕ s λ , ϕ l λ with λ ∈ P/MQ , moreover λ ∈ Λ M zero values: λ ∈ MH s ∨ ∩ Λ M , u ∈ F s ϕ s λ ( u ) = 0 , M λ ∈ MH l ∨ ∩ Λ M , u ∈ F l ϕ l λ ( u ) = 0 , M Corollary ϕ s λ ( u ) , u ∈ F s M , λ ∈ Λ s M ϕ l λ ( u ) , u ∈ F l M , λ ∈ Λ l M J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  81. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Outline 1 Short and long orbit functions Lie groups/Lie algebras Sign homomorphisms S s − and S l − functions 2 Discretization of orbit functions Grids F s M and F l M Grids Λ s M and Λ l M Discrete orthogonality of S s − and S l − functions J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  82. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete orthogonality of S s − and S l − functions x ∈ R n /Q ∨ , the isotropy group Stab( x ) = { w ∈ W | wx = x } the orbit Wx = { wx ∈ R n /Q ∨ | w ∈ W } h x ≡ | Stab( x ) | , ε ( x ) ≡ | Wx | ε ( x ) = | W | h x λ ∈ R n /MQ , the isotropy group Stab ∨ ( λ ) = { w ∈ W | wλ = λ } λ ≡ | Stab ∨ ( λ ) | h ∨ J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  83. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete orthogonality of S s − and S l − functions a scalar product for f, g : F s M → C � � f, g � F s M = ε ( x ) f ( x ) g ( x ) x ∈ F s M a scalar product for f, g : F l M → C � � f, g � F l M = ε ( x ) f ( x ) g ( x ) x ∈ F l M Theorem For λ, λ ′ ∈ Λ s M it holds that M = c | W | M n h ∨ � ϕ s λ , ϕ s λ ′ � F s λ δ λ,λ ′ and for λ, λ ′ ∈ Λ l M it holds that M = c | W | M n h ∨ � ϕ l λ , ϕ l λ ′ � F l λ δ λ,λ ′ . J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  84. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete S s − and S l − transforms interpolating functions I s M , I l M I s � c s λ ϕ s I l � c l λ ϕ l x ∈ R n M ( x ) := λ ( x ) , M ( x ) := λ ( x ) , λ ∈ Λ s λ ∈ Λ l M M the interpolation of f : F M → C : find c s λ (or c l λ ) I s x ∈ F s M ( x ) = f ( x ) , M I l x ∈ F l M ( x ) = f ( x ) , M Proposition � f, ϕ s λ � F s c s = ( c | W | M n h ∨ λ ) − 1 � M ε ( x ) f ( x ) ϕ s λ = λ ( x ) � ϕ s λ , ϕ s λ � F s x ∈ F s M M � f, ϕ l λ � F l c l = ( c | W | M n h ∨ λ ) − 1 � λ = M ε ( x ) f ( x ) ϕ l λ ( x ) � ϕ l λ , ϕ l λ � F l x ∈ F l M M J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

Recommend

Invariance groups of functions, orbit equivalence and group actions Reinhard P oschel

Invariance groups of functions, orbit equivalence and group actions Reinhard P oschel

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Invariance groups of functions, orbit equivalence and group actions Reinhard P oschel Institut f ur Algebra

751 views • 36 slides
The fourth moment of Dirichlet L -functions along a coset and the Weyl bound Ian Petrow ETH Z

The fourth moment of Dirichlet L -functions along a coset and the Weyl bound Ian Petrow ETH Z

The fourth moment of Dirichlet L -functions along a coset and the Weyl bound Ian Petrow ETH Z urich Joint work with Matthew P. Young Ian Petrow (ETH Z urich) The 4th moment and the Weyl bound 1 / 16 The subconvexity problem Given an

897 views • 60 slides
Some classes of generalized boundary triplets, Weyl functions, and local point interactions Seppo

Some classes of generalized boundary triplets, Weyl functions, and local point interactions Seppo

Some classes of generalized boundary triplets, Weyl functions, and local point interactions Seppo Hassi (University of Vaasa) Joint work with M. Malamud and V. Derkach Vienna, Dec. 20, 2016 S. Hassi Generalized boundary triplets, Weyl

388 views • 23 slides
ORBIT Project Overview www.orbit-lab.org ORBIT Overview: Project Rationale Wireless

ORBIT Project Overview www.orbit-lab.org ORBIT Overview: Project Rationale Wireless

ORBIT Project Overview www.orbit-lab.org ORBIT Overview: Project Rationale Wireless testbeds motivated by: cost &amp; time needed to develop experimental prototypes need for reproducible protocol evaluations large-scale system

338 views • 18 slides
Weyl algebra S p g q7 S p g q Deformation of Weyl algebra Problems with Weyl algebra

Weyl algebra S p g q7 S p g q Deformation of Weyl algebra Problems with Weyl algebra

C OMPLETED H OPF ALGEBROIDS D OCTORAL PRESENTATION Martina Stoji c Faculty of Science Department of Mathematics October 20, 2017 I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES 1. I

432 views • 39 slides
Invariance groups of finite functions and orbit equivalence of permutation groups Tam as

Invariance groups of finite functions and orbit equivalence of permutation groups Tam as

Invariance groups of finite functions and orbit equivalence of permutation groups Tam as Waldhauser University of Szeged NSAC 2013 Novi Sad, 7th June 2013 Joint work with Eszter Horv ath, Reinhard P oschel, G eza Makay.

732 views • 72 slides
Generalized Weyl algebras and their global dimension V. V. Bavula 1 Generalized Weyl algebras

Generalized Weyl algebras and their global dimension V. V. Bavula 1 Generalized Weyl algebras

Generalized Weyl algebras and their global dimension V. V. Bavula 1 Generalized Weyl algebras Definition of the generalized Weyl algebras . Let D be a ring, = ( 1 , ..., n ) a set of commuting automorphisms of D , ( i j = j i

295 views • 11 slides
3d dualities and Weyl group symmetry Luca Cassia Milano-Bicocca University Based on work with

3d dualities and Weyl group symmetry Luca Cassia Milano-Bicocca University Based on work with

3d dualities and Weyl group symmetry Luca Cassia Milano-Bicocca University Based on work with A. Amariti, arXiv:18XX.XXXXX Supersymmetric theories, dualities and deformations Bern, 16-18 July 2018 Introduction A good strategy to derive new

420 views • 18 slides
How to blow up orbit closures real good (Orbit closures and rational surfaces) Ted Chinburg

How to blow up orbit closures real good (Orbit closures and rational surfaces) Ted Chinburg

How to blow up orbit closures real good (Orbit closures and rational surfaces) Ted Chinburg November 19, 2012 Joint work with Birge Huisgen-Zimmermann and Frauke Bleher. Question: How to bound the geometry of orbit closures? Set-up: k = k

638 views • 4 slides
SPS Orbit studies Contents: - Motivation - Stabilization of orbit at extraction points - Search

SPS Orbit studies Contents: - Motivation - Stabilization of orbit at extraction points - Search

SPS Orbit studies Contents: - Motivation - Stabilization of orbit at extraction points - Search for orbit drift sources presented by Eliana GIANFELICE (Fermilab and CERN) APC Seminar, Fermilab, November 21, 2013 1/61 &lt; &gt;

730 views • 61 slides
Higher order solution of ODEs arising from DG space semi-discretization of nonstationary

Higher order solution of ODEs arising from DG space semi-discretization of nonstationary

Formulation of the problem Space discretization Time discretization Higher order solution of ODEs arising from DG space semi-discretization of nonstationary convection-diffusion problems Miloslav Vlas ak V t Dolej s Charles

1.09k views • 66 slides
Parallel computations of Grbner bases in the Weyl algebra Something to run on a machine with

Parallel computations of Grbner bases in the Weyl algebra Something to run on a machine with

Weyl algebra Grbner basis Faugres F 4 algorithm Parallel computations of Grbner bases in the Weyl algebra Something to run on a machine with 128 cores Anton Leykin Institute for Mathematics and its Applications, Minneapolis MSRI,

813 views • 62 slides
Discretization and Solution of and Solution of Discretization Convection- -Diffusion Problems

Discretization and Solution of and Solution of Discretization Convection- -Diffusion Problems

Discretization and Solution of and Solution of Discretization Convection- -Diffusion Problems Diffusion Problems Convection Howard Elman University of Maryland Overview 1. The convection-diffusion equation Introduction and examples 2.

908 views • 47 slides
Primer on On-orbit Collision &amp; Debris Generation UN Disarmament Commission Working Group, New

Primer on On-orbit Collision &amp; Debris Generation UN Disarmament Commission Working Group, New

Primer on On-orbit Collision &amp; Debris Generation UN Disarmament Commission Working Group, New York April 10, 2019 Dr. Jaganath Sankaran Assistant Professor, Lyndon B. Johnson (LBJ) School of Public Affairs University of Texas Austin,

429 views • 13 slides
Orbit coherence in permutation groups John R. Britnell Department of Mathematics Imperial

Orbit coherence in permutation groups John R. Britnell Department of Mathematics Imperial

Orbit coherence in permutation groups John R. Britnell Department of Mathematics Imperial College London j.britnell@imperial.ac.uk Groups St Andrews 2013 Joint work with Mark Wildon (RHUL) Orbit partitions Let G be a group of permutations of

565 views • 33 slides
40 years of the Weyl anomaly M. J. Duff Physics Department Imperial College London CFT Beyond

40 years of the Weyl anomaly M. J. Duff Physics Department Imperial College London CFT Beyond

40 years of the Weyl anomaly M. J. Duff Physics Department Imperial College London CFT Beyond Two Dimensions Texas A&amp;M March 2012 1 / 50 Abstract Classically, Weyl invariance S ( g , ) = S ( g , ) under = ( x )

881 views • 50 slides
RIGIDITY OF GROUP ACTIONS II. Orbit Equivalence in Ergodic Theory Alex Furman (University of

RIGIDITY OF GROUP ACTIONS II. Orbit Equivalence in Ergodic Theory Alex Furman (University of

RIGIDITY OF GROUP ACTIONS II. Orbit Equivalence in Ergodic Theory Alex Furman (University of Illinois at Chicago) March 1, 2007 Ergodic Theory of II 1 Group Actions II 1 Systems: discrete countable group ( X , B , ) std

890 views • 70 slides
The Superconformal Index and the Weyl Anomaly Great Lakes Strings 2018, University of Chicago

The Superconformal Index and the Weyl Anomaly Great Lakes Strings 2018, University of Chicago

The Superconformal Index and the Weyl Anomaly Great Lakes Strings 2018, University of Chicago Brian McPeak Leinweber Center for Theoretical Physics University of Michigan Based on work presented in 1804.04155 Conformal Symmetry and Weyl

387 views • 10 slides
Orbit matrices of symmetric designs and related self-dual codes Orbit matrices of symmetric

Orbit matrices of symmetric designs and related self-dual codes Orbit matrices of symmetric

Orbit matrices of symmetric designs and related self-dual codes Orbit matrices of symmetric designs and (a joint work with Dean Crnkovi c) related self-dual codes Sanja Rukavina Introduction sanjar@math.uniri.hr Orbit matrices of

487 views • 29 slides
Global Weyl modules and maximal parabolics of twisted affine Lie algebras Matthew Lee Department

Global Weyl modules and maximal parabolics of twisted affine Lie algebras Matthew Lee Department

Motivation Background Realization of Maximal Parabolic Global Weyl Module A Global Weyl modules and maximal parabolics of twisted affine Lie algebras Matthew Lee Department of Mathematics University of California, Riverside Interactions

606 views • 31 slides
M etal-Insulator Transitions in a model for magnetic Weyl semimetal and graphite under high

M etal-Insulator Transitions in a model for magnetic Weyl semimetal and graphite under high

M etal-Insulator Transitions in a model for magnetic Weyl semimetal and graphite under high magnetic field Disorder-driven quantum phase transition in Weyl fermion semimetal Luo, Xu, Ohtsuki and RS, ArXiv:1710.00572v2, Liu, Ohtsuki and RS,

874 views • 38 slides
Discretization and tropicalization: How are they related? Christer O. Kiselman Member of the

Discretization and tropicalization: How are they related? Christer O. Kiselman Member of the

Discretization and tropicalization: How are they related? Christer O. Kiselman Member of the Reference Group for Mathematics of ISP First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm, Sida Headquarters 2017

1.19k views • 93 slides
Group Based Combinatorial Zeta Functions Fourth International Workshop on Zeta Functions in

Group Based Combinatorial Zeta Functions Fourth International Workshop on Zeta Functions in

Group Based Combinatorial Zeta Functions Fourth International Workshop on Zeta Functions in Algebra and Geometry Bielefeld, June 1, 2017 Winnie Li Pennsylvania State University 1 Selberg zeta functions SL 2 ( R ) acts on H by fractional

299 views • 27 slides
Cosserat Rod with rh- Adaptive Discretization Jiahao Wen 1 , Jiong Chen 1 , Nobuyuki Umetani 2 ,

Cosserat Rod with rh- Adaptive Discretization Jiahao Wen 1 , Jiong Chen 1 , Nobuyuki Umetani 2 ,

Cosserat Rod with rh- Adaptive Discretization Jiahao Wen 1 , Jiong Chen 1 , Nobuyuki Umetani 2 , Hujun Bao 1 , Jin Huang 1 1 Zhejiang University 2 The University of Tokyo Background Discretization Approximate infinite dimensional solution space

639 views • 25 slides

More recommend