A343266 - OEIS

%I #22 Jun 02 2024 11:04:05

%S 7980534952482277785600,81594430454916707328000,

%T 14562708974765215732435968000,548418186007940753739828795801600,

%U 5607126565433818566044216721408000,1000741740604516653587154703585640448000,1197835483167781694195253526401026457600

%N List of dimensions for which there exist 8 or more non-isomorphic irreducible representations of E6.

%C Terms which would appear in A121737 eight times or more if it included repeats.

%C There are infinitely many terms in this sequence; see A181746. For example, 548418186007940753739828795801600 = 2^36 * 7980534952482277785600, and additional dimensions may be obtained by replacing 2^36 with n^36 for any positive integer n.

%D N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.

%D J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.

%H Andy Huchala, <a href="/A343266/b343266.txt">Table of n, a(n) for n = 1..12</a>

%H Andy Huchala, <a href="/A343266/a343266.cpp.txt">C++ Program</a>

%e With fundamental weights as ordered in Bourbaki, the eight irreducible E6 representations with dimension 7980534952482277785600 have highest weights as follows:

%e   [ 1, 5, 2, 5, 0, 7]

%e   [ 7, 5, 0, 5, 2, 1]

%e   [ 5, 7, 2, 3, 2, 1]

%e   [ 1, 7, 2, 3, 2, 5]

%e   [ 1, 5, 6, 1, 0,11]

%e   [11, 5, 0, 1, 6, 1]

%e   [ 4, 5, 3, 1, 0,14]

%e   [14, 5, 0, 1, 3, 4]

%o (Java) // See A181746 and A121737.

%o (C++) // See A181746.

%Y Cf. A181746, A121737.

%K nonn

%O 1,1

%A _Andy Huchala_, Apr 09 2021