# Isomorphism via derivations

### Joshua Maglione

UniversitÃ¤t Bielefeld

jmaglione@math.uni-bielefeld.de

(This talk is being recorded.)

## Joint with...

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}

## A bottleneck example

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}

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

Work, via induction, down characteristic filtration:

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 $$\approx q^{3136}$$.

Maybe can construct nontrivial characteristic subgroups (i.e. fixed by every automorphism).

Use these to produce more by refining a generalized filtration (M. (2017, 2021), Wilson (2013)).

Refine

## 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}.
[,] : 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 thatÂ

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\},

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}

Our Example:

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

$$1$$-dim. densors.

## Efficiency with tiny densors

• "Algorithm" = Las Vegas + factoring oracles in $$\mathbb{Z}, \mathbb{Q}[x]$$.
• "Chevalley type" = derived subalgebra has Chevalley basis.
[\mathfrak{gl}_n, \mathfrak{gl}_n] = \mathfrak{sl}_n
\mathfrak{Der}(\mathbb{F}_p[x]/(x^p))

Theorem (Brooksbank, M, Wilson (2020)).

Let $$K = 6K$$ or $$K/\mathbb{Q}$$ be finite. There exists an algorithm that, given nondegenerate $$s, t : K^a\times K^b \rightarrow K^c$$ with $$\mathfrak{Der}(t)$$ of Chevalley type, $$\mathrm{im}(t)=K^c$$, and $$\dim(\hspace{-0.77mm}|t|\hspace{-0.77mm})=1$$, decides $$s\cong t$$ in $$(a+b+c)^{O(1)}$$ steps.

## $$\exists$$ 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

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} \cong K^{n^2}}
\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.

## Densor is densest

Derivations "satisfy" the polynomial $$x + y - z$$.

t(Xu,v) + t(u, Yv) = Zt(u,v)

The densor space is the $$P$$-closure, for $$P=(x+y-z)$$.

For every ideal $$P$$, there is a corresponding $$P$$-closure.

Theorem (First, M, Wilson (2020)).

For every homogeneous linear ideal $$P\subset K[x,y,z]$$, the densor embeds into the $$P$$-closure, provided the bilinear map is nondegenerate and image = codomain.

Densors are smallest; computed only with linear algebra.

## A family of tiny densors

Let $$L=\mathfrak{sl}_{n+1}(K)$$ be simple, and $$V$$ a simple $$L$$-module.

t : L\times V \rightarrow V
(x, v)\mapsto xv

Finite dimensional $$L$$-modules determined by partitions $$\lambda$$.

Lemma. If $$V\cong V(\lambda)$$, then $$\dim(\hspace{-0.77mm}|t|\hspace{-0.77mm})$$ is the number of "corners" of $$\lambda$$. In particular, $$\dim(\hspace{-0.77mm}|t|\hspace{-0.77mm}) \leq n$$.

\dim(\hspace{-0.77mm}|t|\hspace{-0.77mm}) = 3
\begin{aligned} n &= 5,\\ \lambda &= (5, 5, 3, 3, 1) \end{aligned}

## 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)).

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?

## Derivations find a shortcut

Nilpotent groups may lack characteristic structure.

Derivation-densor generalizes previous isomorphism tests and is the most "optimal."

Groups with tiny densors have large $$\mathrm{Aut}$$ of Lie-type.

Derivation-densor is polynomial time with tiny densors and Chevalley type.

Shows potential to be polynomial time on larger families.