Linearly reductive finite subgroup schemes of $${\text {SL}}(3)$$ SL ( 3 )
- PDF / 975,383 Bytes
- 6 Pages / 439.37 x 666.142 pts Page_size
- 15 Downloads / 198 Views
Linearly reductive finite subgroup schemes of SL(3) Yuval Z. Flicker1,2
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract We determine the linearly reductive finite subgroup schemes G of SL(3,F), namely those all of whose modules are completely reducible, where F is an algebraically closed field of positive characteristic, up to conjugation. This extends work of M. Hashimoto from SL(2,F) to SL(3,F). Keywords Finite group schemes · Finite linear groups · SL(2, F) · SL(3, F) · Linearly reductive algebraic groups · Complete reducibility Mathematics Subject Classification 14G17 · 14G27 · 14G25 · 14E08
1 Introduction Let F be an algebraically closed field of positive characteristic, p. We determine the linearly reductive finite subgroup schemes G of SL(3,F), up to conjugation, in analogy with the work of Hashimoto [3], who considered the same question in the context of SL(2,F), using a description of the connected linear reductive affine algebraic F-group schemes by Sweedler [5], and the classification of the finite subgroups of SL(2,F) (a proof of the statement [3, Theorem 3.2] of the classification in the case of SL(2) is provided by [1]). A modern exposition, with a new proof, of the classification of the finite subgroups of SL(3, F)—on which the present work relies— has recently been given in [2]. By considering linearly reductive (= all G-modules are completely reducible) finite F-subgroup schemes of SL(3), rather than finite Communicated by Mikhail Belolipetsky. Partially supported by Israel Absorption Ministry Kamea B Science grant. I warmly thank IPMU at Tokyo University for hospitality when most of this work was conceived. * Yuval Z. Flicker [email protected] 1
Ariel University, 40700 Ariel, Israel
2
The Ohio State University, Columbus, OH 43210, USA
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
subgroups—which is the case of reduced such group schemes—the restriction that the characteristic p = charF of F not divide the order |G| of the finite group G—is relaxed. It is weakened to a restriction on the characteristic p, thus obtaining new finite subgroup schemes which are not reduced: not coming from finite subgroups of SL(3, F) . It will be interesting to explore algebro-geometric applications, perhaps using [7].
2 Preliminaries Let F be a field. Let X be an F-scheme. An affine algebraic F-group scheme G is called linearly reductive if any G-module is semisimple. Describing the preliminaries in this section, we follow the exposition of [3], where proofs of Lemmas 2.1 (see [3, Lemma 2.2]) and 2.2 (see [3, Lemma 2.5]) below are given. Lemma 2.1 Let 1 → N → G → H → 1 be an exact sequence of affine algebraic F-group schemes. Then G is linearly reductive if and only if H and N are linearly reductive. An element c ≠ 0 in an F-coalgebra C = (C, Δ, 𝜀) is called group-like if Δ(c) = c ⊗ c [4]. Then 𝜀(c) = 1 for such c. The set Y(C) of group-like elements of C is linearly independent. Let H be an F-Hopf algebra. The antipode S acts on Y(H) by S(h)
Data Loading...
