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On the lattice of subgroups of Chevalley groups containing a split maximal torus. (English) Zbl 0857.20023
Let \(\Phi\) be a reduced irreducible root system, \(G=G(\Phi,K)\) be a Chevalley group of type \(\Phi\) over a field \(K\), \(T=T(\Phi,K)\) be a split maximal torus in \(G\). In the papers by Z. I. Borevich (for matrix groups), G. Seitz, N. A. Vavilov et al. it has revealed that in many cases each subgroup \(F\supset T\) is standard. This means that there exists a closed set of roots \(S\), such that \(G(S)\subset F\subset N(S)\), where \(G(S)\) is generated by \(T\) and all root subgroups \(X_a\), \(a\in S\), and \(N(S)=N_G(G(S))\). To find inclusions among \(N(S)\) one has to find inclusion among pairs \((S,X(S))\), where \(X(S)=\{w\in W(\Phi)\mid wS=S\}\).
In the present paper are listed all inclusions \(N(\Delta_1)\leq N(\Delta_2)\) and also all inclusions \(N(\Delta_1)\leq G(\Delta_2)\), where \(\Delta_1\), \(\Delta_2\) ranges over root subsystems in \(\Phi\). The results are summarized in Tables 2-6. Table 1 lists all maximal non-parabolic standard subgroups. Of course these descriptions are given up to conjugacy.

MSC:
20G15 Linear algebraic groups over arbitrary fields
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
17B20 Simple, semisimple, reductive (super)algebras
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