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 A109441 A sequence of Cartan group dimensions, sorted and duplicates deleted: Dim{A_n,B_n,C_n,D_n}={(n + 1)^2 - 1, (2*n + 1)*(2*n + 1 - 1)/2, (2*n)^2, (2*n)*(2*n - 1)/2}. 1

%I

%S 0,1,3,4,6,8,10,15,16,21,24,28,35,36,45,48,55,63,64,66,78,80,91,99,

%T 100,105,120,136,143,144,153,168,171,190,195,196,210,224,231,253,255,

%U 256,276,300,324,325,351,378,400,406,435,465,484,576,676,784,900

%N A sequence of Cartan group dimensions, sorted and duplicates deleted: Dim{A_n,B_n,C_n,D_n}={(n + 1)^2 - 1, (2*n + 1)*(2*n + 1 - 1)/2, (2*n)^2, (2*n)*(2*n - 1)/2}.

%C Result gives new exceptional groups isomorphisms: E_6->D_6 ( 78) E_7 1/2-> D_9 ( 190) E_9(E_11 ?)-> C_10 ( 484) And none for G_2,F_4,E_7,E_8. (14,52,133,248) That reasoning separates the exception groups into two types.

%D R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

%D Todd Rowland, "Dynkin Diagram", http://mathworld.wolfram.com/DynkinDiagram.html

%F A_n->Dim_SL(n+1)=(n+1)^2-1 B_n->Dim_SO(2*n+1)=(2*n+1)*(2*n+1-1)/2 C_n->Dim_SP(2*n)=((2*n)^2 D_n->Dim_SO(2*n)=2*n*(2*n-1)/2 a(n)=Sort[{(n + 1)^2 - 1, (2*n + 1)*(2*n + 1 - 1)/2, (2*n)^2, (2*n)*(2*n - 1)/2}]

%t Union[Flatten[Table[{(n + 1)^2 - 1, (2*n + 1)*(2*n + 1 - 1)/2, (2*n)^2, (2*n)*(2*n - 1)/2}, {n, 0, 15}]]]

%Y Cf. A120722.

%K nonn,uned

%O 1,3

%A _Roger L. Bagula_, Jun 22 2007

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