Isomorphism via derivations

Joshua Maglione

Universität Bielefeld

www.slides.com/joshmaglione/gig2020

jmaglione@math.uni-bielefeld.de

(This talk is being recorded.)

Creative Commons Attribution 4.0 International License

Goal: $G, H\leq \mathrm{GL}_d(\mathbb{F}_q)$ decide if $G\cong H$ in $(d\log q)^{O(1)}$ steps.

\begin{aligned} G &= \langle \mathcal{X}\rangle \cong \begin{bmatrix} I_2 & * & * \\ & I_7 & * \\ & & I_6 \end{bmatrix}, & H &= \langle \mathcal{Y}\rangle \cong \begin{bmatrix} I_3 & * & * \\ & I_8 & * \\ & & I_4 \end{bmatrix}, \end{aligned}

\begin{aligned} G &= \langle \mathcal{X}\rangle \cong \begin{bmatrix} I_2 & * & * \\ & I_7 & * \\ & & I_6 \end{bmatrix}, & H &= \langle \mathcal{Y}\rangle \cong \begin{bmatrix} I_3 & * & * \\ & I_8 & * \\ & & I_4 \end{bmatrix}, \end{aligned}

A bottleneck example

1 \longrightarrow K^{12} \longrightarrow E \longrightarrow K^{56} \longrightarrow 1

1 \longrightarrow K^{12} \longrightarrow E \longrightarrow K^{56} \longrightarrow 1

Related: Are $(2, 7, 6)$ and $(3, 8, 4)$ intrinsic to $G$ and $H$ ?

\begin{bmatrix} I_4 & * \\ & I_6 \end{bmatrix} \cong \begin{bmatrix} I_3 & * \\ & I_8 \end{bmatrix}

\begin{bmatrix} I_4 & * \\ & I_6 \end{bmatrix} \cong \begin{bmatrix} I_3 & * \\ & I_8 \end{bmatrix}

Example: $K = \mathbb{F}_q$ and

Apply Cannon, Holt (2003) and Eick et al. (2002).

Work, via induction, down characteristic filtration:

G = \eta_1 > \eta_2 > \cdots > \eta_{c+1} = 1,

G = \eta_1 > \eta_2 > \cdots > \eta_{c+1} = 1,

where $\eta_{i+1} = \eta_i^p[\eta_i, G]$ .

Decide which $G/\eta_i \overset{\cong}{\longrightarrow} H/\eta_i$ lifts to a $G/\eta_{i+1} \overset{\cong}{\longrightarrow} H/\eta_{i+1}$ .

Our Example: $G/\Phi(G) \cong H/\Phi(H) \cong K^{56}$ .

Search through $\mathrm{GL}_{56}(K)$ , has order $\approx q^{3136}$ .

Bilinear maps and derivations

Each $s, t\in \mathbb{N}$ , defines $\mathbb{F}_p$ -bilinear map:

[,]_{s, t} : \eta_{s}/\eta_{s+1} \times \eta_{t}/\eta_{t+1} \rightarrow \eta_{s+t}/\eta_{s+t+1}.

[,]_{s, t} : \eta_{s}/\eta_{s+1} \times \eta_{t}/\eta_{t+1} \rightarrow \eta_{s+t}/\eta_{s+t+1}.

[,] : U \times V \rightarrow W

[,] : U \times V \rightarrow W

Easier: decide isomorphism of bilinear maps, instead.

Find $(\alpha,\beta, \gamma)\in \mathrm{GL}(U)\times \mathrm{GL}(V)\times \mathrm{GL}(W)$ such that

The Lie algebra of derivations for $t : U\times V \rightarrow W$ is

\mathfrak{Der}(t) = \left\{(X, Y, Z)\in \mathfrak{L} ~\middle|~ \begin{array}{c} \forall u\in U, \forall v\in V, \\ t(Xu, v) + t(u, Yv) = Zt(u, v) \end{array} \right\},

\mathfrak{Der}(t) = \left\{(X, Y, Z)\in \mathfrak{L} ~\middle|~ \begin{array}{c} \forall u\in U, \forall v\in V, \\ t(Xu, v) + t(u, Yv) = Zt(u, v) \end{array} \right\},

where $\mathfrak{L} = \mathfrak{gl}(U)\times \mathfrak{gl}(V) \times \mathfrak{gl}(W)$ .

The densor space (derivation tensor) is

\begin{aligned} (\hspace{-0.77mm}|t|\hspace{-0.77mm}) &= \mathrm{hom}_{\mathfrak{Der}(t)}(U\otimes V, W) \\ &= \left\{ (s : U\times V \rightarrow W) ~\middle|~ \mathfrak{Der}(t) \subseteq \mathfrak{Der}(s) \right\} \end{aligned}

\begin{aligned} (\hspace{-0.77mm}|t|\hspace{-0.77mm}) &= \mathrm{hom}_{\mathfrak{Der}(t)}(U\otimes V, W) \\ &= \left\{ (s : U\times V \rightarrow W) ~\middle|~ \mathfrak{Der}(t) \subseteq \mathfrak{Der}(s) \right\} \end{aligned}

Our Example:

\mathfrak{Der}([,]_G) \cong \mathfrak{sl}_2 \oplus \mathfrak{sl}_7\oplus \mathfrak{sl}_6\oplus K^2

\mathfrak{Der}([,]_G) \cong \mathfrak{sl}_2 \oplus \mathfrak{sl}_7\oplus \mathfrak{sl}_6\oplus K^2

\mathfrak{Der}([,]_H) \cong \mathfrak{sl}_3 \oplus \mathfrak{sl}_8\oplus \mathfrak{sl}_4\oplus K^2

\mathfrak{Der}([,]_H) \cong \mathfrak{sl}_3 \oplus \mathfrak{sl}_8\oplus \mathfrak{sl}_4\oplus K^2

\left.\begin{array}{c} \\ \\ \end{array} \right\}

\left.\begin{array}{c} \\ \\ \end{array} \right\}

$1$ -dim. densors.

$\exists$ more associative way?

Example: $\mathfrak{sl}_n$ acts on $K^n$ . Determines $t: \mathfrak{sl}_n\times K^n \rightarrow K^n$ .

U\times V \times W^* \rightarrow K

U\times V \times W^* \rightarrow K

Usual suspects do not help, yet $\dim(\hspace{-0.77mm}|t|\hspace{-0.77mm}) = 1$ .

{\color{blue} \approx K^{n^3}}

{\color{blue} \approx K^{n^3}}

{\color{blue} \approx K^{n^3}}

{\color{blue} \approx K^{n^3}}

{\color{blue} \cong K^{n^2}}

{\color{blue} \cong K^{n^2}}

\begin{aligned} U &= \mathfrak{sl}_n \\ V &= K^n \\ W &= K^n \end{aligned}

\begin{aligned} U &= \mathfrak{sl}_n \\ V &= K^n \\ W &= K^n \end{aligned}

But sometimes associative algebras are enough.

Core idea of isomorphism test based on:

Theorem (Brooksbank, M, Wilson (2017)).

There exists an algorithm that, given class 2 $p$ -groups $G, H$ of exponent $p$ with $G'\cong H' \cong \mathbb{F}_p^2$ , decides if $G\cong H$ in $(p + \log|G|)^{O(1)}$ steps.

Derivation-densor method

Automorphism version: compute stabilizer instead.

The algorithm works regardless of our hypotheses:

Compute $\mathfrak{Der}$ and $(\hspace{-0.77mm}|\cdot|\hspace{-0.77mm})$ for both $s, t$ .

Find invertible $\varphi$ such that $\varphi\mathfrak{Der}(s)\varphi^{-1} = \mathfrak{Der}(t)$ .

Construct gens for normalizer $N := \mathrm{N}(\mathfrak{Der}(t))$ .

Decide if $s^{\varphi}$ and $t$ in same $N$ -orbit in $(\hspace{-0.77mm}|t|\hspace{-0.77mm})$ .

1.

2.

3.

4.

Much work to find the conjugating element $\varphi$ .

Tiny densor $\implies$ $U, V, W$ simple $\mathfrak{Der}$ -mods.

Decompose: $\mathfrak{Der}(t) = M_0\oplus M_1\oplus \cdots \oplus M_r$ (Ivanyos et al. (2012)).

Decide conjugacy on abelians (Brooksbank, Wilson (2015)).

Apply de Graaf (2000), Ryba (2007), Magaard, (R. A.) Wilson (2012) to get Chevalley basis.

Tensor decompose $U, V, W$ .

Decide conjugacy of simple Lie modules of simple Lie algebras (Grochow (2012)).

Tiny densor $\implies$ gens for $\mathrm{N}(\mathfrak{Der})$ not required.

When $K$ is finite, can construct automorphisms.

Derivation-densor method applies more generally.

Question. Which conditions can be relaxed without losing polynomial time?

Warning: Grochow proved general conjugacy of Lie modules is as hard as Graph Isomorphism (2012).

$\dim (\hspace{-0.77mm}|t|\hspace{-0.77mm})\geq 2$ ?

Non-Chevalley simples?

Isomorphism via derivations

Joshua Maglione

Joint with...

A bottleneck example

Not always possible

Bilinear maps and derivations

Efficiency with tiny densors

$\exists$ more associative way?

Densor is densest

A family of tiny densors

Derivation-densor method

Derivations find a shortcut