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Kostant’s conjecture holds for \(E_ 7\): \(L_ 2(37)<E_ 7(\mathbb{C})\). (English) Zbl 0839.20023

Let \(G\) be an adjoint group of Lie-type over \(\mathbb{C}\) with Coxeter number \(h\). Kostant’s conjecture runs as follows: if \(2h+1\) is a prime power, then \(L_2 (2h+1) <G\) [see A. M. Cohen and R. L. Griess, Proc. Symp. Pure Math. 47, 367-405 (1987; Zbl 0654.22005)]. Taking the known results into account, to settle Kostant’s conjecture, it remains to determine whether \(L_2 (37)\) is contained in \(E_7 (\mathbb{C})\) and whether \(L_2(61)\) is contained in \(E_8 (\mathbb{C})\). In this note the authors report on their computer research, which establishes the conjecture for \(E_7\).

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20E07 Subgroup theorems; subgroup growth

Citations:

Zbl 0654.22005
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