Overgroups of $E(m,R)⊗E(n,R)$ I. Levels and normalizers

Abstract

The third author acknowledges support of the RFBR project 10-01-92651 "Higher composition laws, algebraic K-theory, and exceptional groups".

Overgroups of exterior powers of an elementary group. I. Levels and normalizers

“…In the general case, this [partially ordered] set of ideals forms the a net of ideals (due to Zenon Borevich). Note that in the case of not-unique ideal parametrization, even the notion of relative elementary group is far more complex and depends on the Chevalley group (for instance, see [13]), let alone formulations of the Main Theorems.…”

“…• for exceptional groups of Lie type E 6 , E 7 , E 8 , F 4 by Nikolai Vavilov and Alexander Luzgarev in [9,10,11,12]; • for tensor product of general linear groups GL m ⊗ GL n the problem was partially solved by Alexey Ananievskii, Nikolai Vavilov, and Sergey Sinchuk in the joint paper [13]. We highly recommend the surveys [14,15,16], which contain necessary preliminaries, complete history, and known results of the problem.…”

Overgroups of exterior powers of an elementary group. Levels

“…This case corresponds to the Aschbacher class Ë consisting of almost simple groups in certain absolutely irreducible representations. Morally, the paper is a continuation of a series of papers by the St. Petersburg school on subgroups in classical groups over a commutative ring, see [13,[16][17][18][19][20][21][22][23][24][25][26][27][28].…”

“…There are a lot of nontrivial relationships between the ideals. So even the notion of a relative elementary group is far more complicated and depends on a Chevalley group (for instance, see [28]), let alone formulations of the Main Theorems. The authors work in this direction and hope this problem would be solved in the near future.…”

Overgroups of elementary groups in polyvector representations

“…The plausibility of the results for m E n (R) seems obvious due to the close connection of the m-th vector representation and the tensor product of elementary linear groups. Recall that all overgroups of E k (R) ⊗ E l (R) are parametrized by a triple of ideals (A, B, C) in the ring R, see [1]. Moreover, in the important particular case m = 2, n = 4 the problem was completely solved in the paper [14] devoted to elementary orthogonal groups 1 .…”