[Paper Review] On extensions for Ree groups of type F_4

David I. Stewart|arXiv (Cornell University)|Apr 9, 2013

Finite Group Theory Research5 references3 citations

TL;DR

This paper extends the cohomological results of BNP06 to Ree groups of type $F_4$ in characteristic 2, proving that for large enough Ree groups ${}^2F_4(q)$, self-extensions of simple modules vanish generically and that $\mathrm{H}^1(G(\sigma), L)$ for $G(\sigma)$-modules $L$ can be identified with $\mathrm{H}^1(G, M)$ for some $G$-module $M$. The key contribution is establishing analogues of BNP06's theorems on 1-cohomology and extensions for these previously unaddressed finite groups of Lie type.

ABSTRACT

Let k be an algebraically closed field of characteristic p=2. Let G=F_4 be simply connected over k and let σ:G o G be an endomorphism such that the fixed point set G(σ) is a Ree group. We show, using the methods of Bendel--Nakano--Pillen, that self-extensions of simple kG(σ)-modules vanish generically and that for all but the first few Ree groups of type F_4, the 1-cohomology for G(σ) with coefficients in simple kG-modules can be identified with the 1-cohomology for G with coefficients in (possibly different) simple G-modules.

Motivation & Objective

• To close the gap in cohomological results for Ree groups of type $F_4$ in characteristic 2, which were excluded in prior work by BNP06.
• To establish analogues of BNP06's Theorems 5.4, 5.5, and 5.6 for the finite Ree groups ${}^2F_4(q)$.
• To show that for all but finitely many Ree groups of type $F_4$, the $1$-cohomology of simple modules over $G(\sigma)$ is isomorphic to the $1$-cohomology of a (possibly different) simple $G$-module.
• To prove that self-extensions of simple $kG(\sigma)$-modules vanish generically for these groups.

Proposed method

• Use spectral sequences, particularly the LHS spectral sequence, to relate $\mathrm{Ext}^1_{G(\sigma)}(L(\lambda), L(\mu))$ to $\mathrm{Ext}^1_G$-terms via intermediate Frobenius kernels $G_{r/2}$.
• Apply the $\tau$-adic expansion of weights $\lambda = \sum_{i=0}^r (\tau^*)^i \lambda_{i/2}$ to decompose modules and reduce cohomological computations.
• Use the module $\mathcal{G}_{\Omega}(k)$, a truncation of the induced module $\mathcal{G}(k) = \mathrm{Ind}_{G(\sigma)}^G(k)$, to control cohomological filtrations and relate $G(\sigma)$-extensions to $G$-extensions.
• Leverage Sin’s results on $\mathrm{Ext}^1$ for $G_{1/2}$-modules and the fact that $\mathrm{Ext}^1_{G_{1/2}}(L(\lambda_1), L(\lambda_1)) = 0$ to eliminate terms in spectral sequences.
• Use the Steinberg tensor product theorem and twist functors $[\sigma]$ to relate $G(\sigma)$-modules to $G$-modules via $L(\tilde{\lambda}) \cong L(\lambda)^{[n/2]}$.
• Apply Proposition 3.4 to reduce $\mathrm{Ext}^1_{G(\sigma)}$ to a sum over $\nu \in \Gamma'$, and show the sum vanishes when $\lambda_{i/2} \neq \mu_{i/2}$ by analyzing $\mathrm{Hom}_G$-terms and using $\mathrm{Ext}^1_{G_{r/2-t}}(L(\lambda_{1}), L(\lambda_{1})) = 0$.

Experimental results

Research questions

• RQ1 Can the vanishing of self-extensions for $G(\sigma)$-modules be established generically for Ree groups of type $F_4$ in characteristic 2?
• RQ2 Is the $1$-cohomology of a simple $kG(\sigma)$-module isomorphic to the $1$-cohomology of a simple $G$-module for all but finitely many $q$?
• RQ3 Can the cohomological comparison between $G(\sigma)$ and $G$ be extended to $\mathrm{Ext}^1$ between arbitrary simple modules, not just self-extensions?
• RQ4 How can spectral sequences involving $G_{r/2}$-kernels be used to reduce $\mathrm{Ext}^1_{G(\sigma)}$ computations to classical Frobenius kernels $G_r$?

Key findings

• For all but finitely many Ree groups ${}^2F_4(q)$ in characteristic 2, the $1$-cohomology $\mathrm{H}^1(G(\sigma), L)$ for a simple $kG(\sigma)$-module $L$ is isomorphic to $\mathrm{H}^1(G, M)$ for some simple $G$-module $M$, provided $q = 2^s$ with $s \geq 10$.
• The self-extensions $\mathrm{Ext}^1_{G(\sigma)}(L(\lambda), L(\lambda))$ vanish generically for $G(\sigma)$-modules $L(\lambda)$, as shown via spectral sequence arguments and the vanishing of $\mathrm{Ext}^1_{G_{1/2}}(L(\lambda_1), L(\lambda_1))$.
• The $\mathrm{Ext}^1_{G(\sigma)}(L(\lambda), L(\mu))$ is isomorphic to $\mathrm{Ext}^1_G(L(\tilde{\lambda}), L(\tilde{\mu}))$ for a modified weight $\tilde{\lambda}$, constructed via the $\tau$-adic expansion and a shift by $n/2$ in the index, when $s \geq 10$.
• The spectral sequence reduction relies on the vanishing of $\mathrm{Ext}^1_{G_{r/2-t}}(L(\lambda_1), L(\lambda_1))$ and $\mathrm{Hom}_{G_{1/2}}$-terms, which follows from Sin’s results and standard cohomological vanishing.
• The truncation $\mathcal{G}_{\Omega}(k)$ of the induced module $\mathcal{G}(k)$ has only finitely many sections, one for each $\nu \in \Gamma = \{\nu \in X^+ \mid \langle \nu, \alpha_0^\vee \rangle < 2(h-1)\}$, and controls the cohomological filtration.
• The proof uses the fact that $L(\tilde{\lambda}) \cong L(\lambda)^{[n/2]}$ as $G(\sigma)$-modules, and that $\mathrm{Ext}^1_{G(\sigma)}(L(\tilde{\lambda}), L(\tilde{\mu}))$ injects into $\mathrm{Ext}^1_G(L(\tilde{\lambda}), L(\tilde{\mu}))$ via the automorphism $\tau$, enabling comparison with $G$-cohomology.