[Paper Review] Polar spaces and embeddings of classical groups

Nick Gill|ArXiv.org|Mar 15, 2006

Finite Group Theory Research4 references17 citations

TL;DR

This paper constructs embeddings of classical isometry groups from polar spaces over finite fields $GF(q^w)$ into larger classical groups over $GF(q)$ via field trace maps. By composing forms with an $F$-linear functional $L: K \to F$, it classifies the resulting forms' types (unitary, orthogonal, symplectic, or degenerate) and determines when the induced group embeddings are maximal, particularly in the finite field case, extending Kleidman and Liebeck's work on Aschbacher's class $\mathcal{C}_3$. The key contribution is a complete classification of the isometry group embeddings induced by such field extension constructions.

ABSTRACT

Given polar spaces $(V,β)$ and $(V,Q)$ where $V$ is a vector space over a field $K$, $β$ a reflexive sesquilinear form and $Q$ a quadratic form, we have associated classical isometry groups. Given a subfield $F$ of $K$ and an $F$-linear function $L:K o F$ we can define new spaces $(V,Lβ)$ and $(V,LQ)$ which are polar spaces over $F$. The construction so described gives an embedding of the isometry groups of $(V,β)$ and $(V,Q)$ into the isometry groups of $(V,Lβ)$ and $(V,LQ)$. In the finite field case under certain added restrictions these subgroups are maximal and form the so called {\it field extension subgroups} of Aschbacher's class $\curlyc{3}$ \cite{aschbacher}. We give precise descriptions of the polar spaces so defined and their associated isometry group embeddings. In the finite field case our results give extra detail to the account of maximal field extension subgroups given by Kleidman and Liebeck \cite[p112]{kl}.

Motivation & Objective

• To systematically construct and classify embeddings of classical isometry groups from polar spaces over $GF(q^w)$ into classical groups over $GF(q)$ using field trace maps.
• To determine the type (symmetric, alternating, hermitian, or degenerate) of the composed forms $L\beta$ and $LQ$ under various field and form conditions.
• To identify when the induced group embeddings are maximal subgroups in Aschbacher's class $\mathcal{C}_3$, particularly in the finite field case.
• To resolve open cases in the degeneracy of $LQ$ when $char(K) = 2$, $|K| + \dim_K V$ is infinite, and $f_Q$ is degenerate.
• To provide precise conditions under which the isometry groups of $L\beta$ and $LQ$ are of type $O^{\pm}(Aw,q)$, $Sp(Aw,q)$, or $U(Aw,q)$, based on $\alpha$, $q$, $w$, and $A$.

Proposed method

• Define a field trace map $L: K \to F$ as $L(x) = \operatorname{Tr}_{K/F} (\alpha x)$ for $\alpha \in K^*$, where $K = GF(q^w)$ and $F = GF(q)$, to induce forms over $F$.
• Compose a reflexive $\sigma$-sesquilinear form $\beta$ or quadratic form $Q$ on $V$ with $L$ to obtain $L\beta: V \times V \to F$ and $LQ: V \to F$, treating $V$ as a vector space over $F$.
• Analyze the degeneracy of $L\beta$ and $LQ$ using properties of the trace map and the structure of $K/F$, particularly when $K$ is a finite field of characteristic 2.
• Classify the type of $L\beta$ and $LQ$ based on the original form type ($\beta$ hermitian, symmetric, alternating) and conditions on $\sigma(\alpha)$, $q$, $w$, and $A$, using trace identities and discriminant analysis.
• Use the theory of quadratic forms and polar spaces over finite fields to determine the Witt index and type ($O^+$, $O^-$, $Sp$, etc.) of the resulting forms over $GF(q)$.
• Apply results from Lam and Taylor on non-degeneracy and radical structure to prove that $LQ$ is degenerate when $char(K) = 2$, $\dim_K V$ is odd, and $Q$ is non-degenerate.

Experimental results

Research questions

• RQ1 Under what conditions is the composed form $L\beta$ non-degenerate when $\beta$ is a reflexive $\sigma$-sesquilinear form and $L: K \to F$ is $F$-linear?
• RQ2 When is the composed quadratic form $LQ$ non-degenerate, particularly in characteristic 2 where the polar form of $Q$ may be degenerate?
• RQ3 What is the type (orthogonal, symplectic, unitary, or atypical) of the form $L\beta$ or $LQ$ when $\beta$ is hermitian and $L$ is a trace map?
• RQ4 How do the isometry groups of $L\beta$ and $LQ$ embed into classical groups over $GF(q)$, and when are these embeddings maximal subgroups in Aschbacher's class $\mathcal{C}_3$?
• RQ5 What are the precise conditions on $\alpha$, $q$, $w$, and $A$ that determine whether the isometry group of $L\beta$ is $O^{+}(Aw,q)$, $O^{-}(Aw,q)$, $Sp(Aw,q)$, or $U(Aw,q)$?

Key findings

• The composed form $L\beta$ is non-degenerate if and only if $\beta$ is non-degenerate and $L \neq 0$, regardless of the characteristic of $K$.
• When $char(K) \neq 2$ or $K = GF(2^h)$ with $\dim_K V$ even, $LQ$ is non-degenerate if and only if $Q$ is non-degenerate and $L \neq 0$.
• When $K = GF(2^h)$ and $\dim_K V$ is odd, $LQ$ is degenerate even if $Q$ is non-degenerate, due to the radical of $f_Q$ being nontrivial and preserved under $L$.
• For a hermitian form $\beta$ over $GF(q^w)$, the composed form $L\beta$ is alternating if $w$ is even, $q$ is even, and $\sigma(\alpha) = \alpha$, leading to an embedding $U(A,q^w) \leq Sp(Aw,q)$.
• When $w$ is even, $q$ is odd, and $\sigma(\alpha) = \alpha$, the type of $L\beta$ is $O^+$ if $A$ is even and $O^-$ if $A$ is odd, giving embeddings $U(A,q^w) \leq O^{\pm}(Aw,q)$.
• The paper provides a complete classification of the isometry group embeddings induced by trace maps, showing that $O(A,q^w) \leq O^{\pm}(Aw,q)$ or $Sp(Aw,q)$ depending on $w$, $q$, and the trace parameter $\alpha$, with explicit conditions on $\alpha\gamma$ when $Q(v) = \gamma v^2$ on the germ $U$.