Asymptotics of commuting probabilities in reductive algebraic groups

Let \(G\) be an algebraic group. For \(d\geq 1\), we define the commuting probabilities \(cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}\), where \(\mathfrak C_d(G)\) is the variety of commuting \(d\)-tuples in \(G\). We prove that for a reductive group \(G\) when \(d\) is large, \(cp_d(G)\sim \fr...

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Bibliographic Details
Published in:arXiv.org 2021-05
Main Authors: Garge, Shripad M, Uday Bhaskar Sharma, Singh, Anupam
Format: Article
Language:English
Subjects:
Algebra
Commuting
Fields (mathematics)
Group theory
Subgroups
Online Access:Get full text
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Summary:Let \(G\) be an algebraic group. For \(d\geq 1\), we define the commuting probabilities \(cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}\), where \(\mathfrak C_d(G)\) is the variety of commuting \(d\)-tuples in \(G\). We prove that for a reductive group \(G\) when \(d\) is large, \(cp_d(G)\sim \frac{\alpha}{n}\) where \(n=\dim(G)\), and \(\alpha\) is the maximal dimension of an Abelian subgroup of \(G\). For a finite reductive group \(G\) defined over the field \(\mathbb F_q\), we show that \(cp_{d+1}(G(\mathbb F_q))\sim q^{(\alpha-n)d}\), and give several examples.
ISSN:2331-8422