On base sizes for algebraic groups. (English) Zbl 1445.20002
The authors outline their investigations:
“Let \(G\) be a (closed) connected affine algebraic group over an algebraically closed field \(K\) of characteristic \(p \geq 0\). Let \(\Omega\) be a faithful transitive \(G\)-variety with point stabilizer \(H\), so we may identify \(\Omega\) with the coset variety \(G/H\). We define three base-related measures that arise naturally in this context:
(i) The exact base size, denoted \(b(G, H )\), is the smallest integer \(c\) such that \(\Omega\) contains \(c\) points with trivial pointwise stabilizer.
(ii) The connected base size, denoted \(b^0(G,H)\), is the smallest integer \(c\) such that \(\Omega\) contains \(c\) points whose pointwise stabilizer has trivial connected component, i.e. the pointwise stabilizer is finite.
(iii) The generic base size, denoted \(b^1(G,H)\), is the smallest integer \(c\) such that the product variety \(\Omega^c = \Omega\times \cdots \times \Omega\) (\(c\) factors) contains a non-empty open subvariety \(\Lambda\) and every \(c\)-tuple in \(\Lambda\) is a base for \(G\).
Evidently, we have \[b^0(G,H) \leq b(G,H) \leq b^1(G,H).\] Our ultimate goal is to determine these base-related measures for all simple algebraic groups \(G\) and all closed maximal subgroups \(H\) of \(G\) (that is, for all primitive actions of simple algebraic groups). Indeed, we essentially achieve this goal by computing these quantities in almost every case. In the handful of exceptional cases, we give a very narrow range for the possible values.”
The authors give a list of 73 references. They refer extensively to previous research. They include a number of lists of results such as: some subspace actions; values of \(b\) in Theorem 5(ii); \(G\) exceptional group, \(H\) parabolic subgroup; values of \(b\) in Theorem 7(ii); involutions inverting maximal tori, \(p \neq 2\); non-subspace involution-type subgroups; values of \(b\) in Theorem 3.13(ii); the \(\mathcal{C}_i \) collections; some maximal non-parabolic subgroups of exceptional groups; \(G\) exceptional group, dim \(G/P_i\); \(D = C_G(x)\), \(x\) semisimple, dim \(x^G \leq 100\); the fusion of unipotent classes, \(A_2G_2 < E_6\).
“Let \(G\) be a (closed) connected affine algebraic group over an algebraically closed field \(K\) of characteristic \(p \geq 0\). Let \(\Omega\) be a faithful transitive \(G\)-variety with point stabilizer \(H\), so we may identify \(\Omega\) with the coset variety \(G/H\). We define three base-related measures that arise naturally in this context:
(i) The exact base size, denoted \(b(G, H )\), is the smallest integer \(c\) such that \(\Omega\) contains \(c\) points with trivial pointwise stabilizer.
(ii) The connected base size, denoted \(b^0(G,H)\), is the smallest integer \(c\) such that \(\Omega\) contains \(c\) points whose pointwise stabilizer has trivial connected component, i.e. the pointwise stabilizer is finite.
(iii) The generic base size, denoted \(b^1(G,H)\), is the smallest integer \(c\) such that the product variety \(\Omega^c = \Omega\times \cdots \times \Omega\) (\(c\) factors) contains a non-empty open subvariety \(\Lambda\) and every \(c\)-tuple in \(\Lambda\) is a base for \(G\).
Evidently, we have \[b^0(G,H) \leq b(G,H) \leq b^1(G,H).\] Our ultimate goal is to determine these base-related measures for all simple algebraic groups \(G\) and all closed maximal subgroups \(H\) of \(G\) (that is, for all primitive actions of simple algebraic groups). Indeed, we essentially achieve this goal by computing these quantities in almost every case. In the handful of exceptional cases, we give a very narrow range for the possible values.”
The authors give a list of 73 references. They refer extensively to previous research. They include a number of lists of results such as: some subspace actions; values of \(b\) in Theorem 5(ii); \(G\) exceptional group, \(H\) parabolic subgroup; values of \(b\) in Theorem 7(ii); involutions inverting maximal tori, \(p \neq 2\); non-subspace involution-type subgroups; values of \(b\) in Theorem 3.13(ii); the \(\mathcal{C}_i \) collections; some maximal non-parabolic subgroups of exceptional groups; \(G\) exceptional group, dim \(G/P_i\); \(D = C_G(x)\), \(x\) semisimple, dim \(x^G \leq 100\); the fusion of unipotent classes, \(A_2G_2 < E_6\).
Reviewer: Erich W. Ellers (Toronto)
MSC:
| 20B15 | Primitive groups |
| 20G15 | Linear algebraic groups over arbitrary fields |
| 20G41 | Exceptional groups |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20G40 | Linear algebraic groups over finite fields |
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