This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339250 #35 May 31 2024 14:50:36 %S A339250 27,351,1728,3003,5824,7371,7722,17550,19305,34398,46332,51975,54054, %T A339250 61425,78975,100386,112320,146432,252252,314496,359424,371800,386100, %U A339250 393822,412776,442442,459459 %N A339250 List of dimensions for which there exist several non-isomorphic irreducible representations of E6. %C A339250 Terms which would be repeated in A121737. %C A339250 There are infinitely many terms in this sequence; see A181746. %C A339250 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 A339250 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 A339250 N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002. %D A339250 J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997. %H A339250 Andy Huchala, <a href="/A339250/b339250.txt">Table of n, a(n) for n = 1..20000</a> %H A339250 Andy Huchala, <a href="/A339250/a339250_2.java.txt">Java program</a> %H A339250 Wikipedia, <a href="https://en.wikipedia.org/wiki/E6_(mathematics)">E6 (mathematics)</a> %e A339250 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 A339250 (Java) // See Links section above and in A181746. %o A339250 (C++) // See Links section of A181746. %Y A339250 Cf. A181746, A121737. %K A339250 nonn %O A339250 1,1 %A A339250 _Andy Huchala_, Apr 02 2021