A339250 - OEIS

A339250

List of dimensions for which there exist several non-isomorphic irreducible representations of E6.

%I #35 May 31 2024 14:50:36

%S 27,351,1728,3003,5824,7371,7722,17550,19305,34398,46332,51975,54054,

%T 61425,78975,100386,112320,146432,252252,314496,359424,371800,386100,

%U 393822,412776,442442,459459

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

%C Terms which would be repeated in A121737.

%C There are infinitely many terms in this sequence; see A181746.

%C By symmetry of the Dynkin diagram, with fundamental weights numbered as in Bourbaki there is a duality of highest weights [1,0,0,0,0,0] and [0,0,0,0,0,1]. Similarly, there is a duality of highest weights [0,0,0,0,1,0] and [0,0,1,0,0,0]. Note that E6 is the only exceptional Lie algebra with such a duality. However this duality is not responsible for all pairs of non-isomorphic irreducible E6 representations of equal dimension--see example.

%C There are 6 non-isomorphic irreducible E6 representations of dimension 7183313280, and 8 non-isomorphic irreducible E6 representations of dimension 7980534952482277785600. Both dimensions are minimal with respect to that property. I do not know if such dimensions exist for 9 or more irreducible representations.

%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="/A339250/b339250.txt">Table of n, a(n) for n = 1..20000</a>

%H Andy Huchala, <a href="/A339250/a339250_2.java.txt">Java program</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/E6_(mathematics)">E6 (mathematics)</a>

%e With the fundamental weights numbered as in Bourbaki, the irreducible E6-modules with highest weights [1,0,0,0,0,0] and [0,0,0,0,0,1] both have dimension 77. The vectors [0,0,0,0,1,0], [0,0,1,0,0,0], [2,0,0,0,0,0], and [0,0,0,0,0,2] are the four highest weights which correspond to irreducible representations of dimension 351.

%o (Java) // See Links section above and in A181746.

%o (C++) // See Links section of A181746.

%Y Cf. A181746, A121737.

%K nonn

%O 1,1

%A _Andy Huchala_, Apr 02 2021