Complete Commutator-Word Tomography of Alternating Pencils

1 Introduction

Let $p$ be an odd prime and let $k=\mathbb F_p$. A special finite $p$-group $G$ of nilpotency class two and exponent $p$ has

and its commutator induces an alternating bilinear map

Conversely, every such alternating map gives a class-two exponent-$p$ group

with multiplication

The commutator is

When $\dim W=2$, the commutator tensor is an alternating pencil. After choosing a basis of $W$, write

where $A,B$ are alternating forms on $V$. The corresponding matrix pencil is

Alternating pencils over fields of characteristic different from $2$ have a classical canonical decomposition into regular elementary-divisor blocks and singular Kronecker blocks. This is the skew-symmetric version of Kronecker pencil theory, and it appears in canonical-form treatments of pairs of forms and pairs of skew-symmetric matrices under congruence.

The purpose of this paper is to show that the entire canonical datum of an alternating pencil is observable from commutator-word distributions.

The guiding mechanism is Fourier-tagged word tomography. Given rectangular coefficient matrices $C,D$, one forms central commutator outputs whose Fourier coefficients are

Thus commutator-word distributions can be used as rank probes for the pencil.

The singular Kronecker blocks are detected by rectangular chain probes. The regular elementary divisors are detected by closed-point Artin probes, which evaluate the pencil at infinitesimal neighbourhoods of closed points of $\mathbb P^1$. Together these probes recover the full canonical datum.

1.1 Main theorem

The main theorem is the following.

Theorem 1.1 (Theorem A -- Complete commutator-word tomography of alternating pencils).

Fix a finite field $k=\mathbb F_p$, $p\neq2$, and an integer $N$. There exists an explicit finite family of Fourier-tagged commutator-word distributions such that, for every alternating pencil

\[ \beta:\Lambda^2_k V\to k^2, \qquad \dim V=N, \]

these distributions determine the projective congruence class of $\beta$.

Equivalently, they determine:

• the multiplicities of all singular Kronecker blocks;

• the multiplicities of all regular elementary divisors ($\xi,e$), up to the natural $\operatorname{PGL}_2(k)$-action on $\mathbb P^1$.

Consequently, for special class-two exponent-$p$ groups $G_\beta$ with

\[ \dim_{\mathbb F_p}G'=2, \]

the same finite family determines the group isomorphism class.

1.2 Scope

The theorem is exact and finite. It is not a sampling theorem and not an optimized isomorphism algorithm. The orbit-coding step used to pass from a chosen target basis to an intrinsic projective invariant is finite but not intended to be computationally efficient.

The contribution is instead structural: the complete skew-symmetric Kronecker canonical data of an alternating commutator pencil is observable from explicitly designed commutator-word distributions.

2 Class-two exponent-$p$ groups and alternating pencils

Let $k=\mathbb F_p$, $p\neq2$, and let

be alternating.

Define

with multiplication

Then $G_\beta$ is a group of exponent $p$ and nilpotency class at most two. Its commutator is

If $\beta$ is surjective and has zero radical,

then

so $G_\beta$ is special.

Conversely, every special class-two exponent-$p$ group arises this way.

Two tensors

are pseudo-isometric if there are vector-space isomorphisms

such that

for all $v,v'\in V$. For special class-two exponent-$p$ groups, group isomorphism is equivalent to pseudo-isometry of the commutator tensor.

Throughout the main body of this paper,

After choosing a basis of $W$, we write

where

Changing the basis of $W$ acts on the projective parameter line by $\operatorname{PGL}_2(k)$.

3 Fourier-tagged commutator distributions

Let

For variables

in $G_\beta$, define two central outputs

where $\bar x_j,\bar y_i$ denote the images in $V=G/Z(G)$.

Thus

is a random variable in

Let

be its probability distribution.

Let

be a nontrivial additive character. For

define the Fourier coefficient

For $\ell\in W^*$, write

If $W=k e_A\oplus k e_B$ and

then

Lemma 3.1 (Bilinear character average).

Let

\[ B_0:U_1\times U_2\to k \]

be a bilinear form between finite-dimensional $k$-vector spaces. Then

\[ \mathbb E_{u\in U_1,\ v\in U_2} \psi(B_0(u,v)) = p^{-\operatorname{rank}B_0}. \]

Theorem 3.2 (Rectangular Fourier-tagged moment formula).

For all

\[ C,D\in M_{r\times s}(k) \]

and all

\[ \alpha,\gamma\in W^*, \]

one has

\[ \boxed{ \widehat P_{C,D}(\alpha,\gamma) = p^{-\operatorname{rank}(C\otimes A_\alpha+D\otimes A_\gamma)}. } \]

Here

\[ C\otimes A_\alpha+D\otimes A_\gamma \]

is regarded as a bilinear form on

\[ (k^s\otimes V)\times(k^r\otimes V). \]

In particular, in the chosen coordinates on $W=k^2$,

\[ \boxed{ \widehat P_{C,D}(e_A^\circ,e_B^\circ) = p^{-\operatorname{rank}(C\otimes A+D\otimes B)}. } \]

This is the basic interface between commutator-word distributions and the linear algebra of matrix pencils.

4 Canonical data of alternating pencils

We recall the canonical data used below.

A skew-symmetric pencil

over a field of characteristic not (2) decomposes as a direct sum of two types of indecomposable blocks.

4.1 Regular blocks

The regular part is described by elementary divisors

where $\xi$ is a closed point of $\mathbb P^1_k$ and $e\ge1$. We encode the regular part by a decorated divisor

A block $(\xi,e)$ contributes dimension

to $V$.

4.2 Singular Kronecker blocks

The singular part is described by minimal indices

Let $m_e$ be the multiplicity of the singular Kronecker block of index $e$. Such a block has dimension

The full canonical data is

up to the natural $\operatorname{PGL}_2(k)$-action on the parameter line $\mathbb P^1$.

The existence and uniqueness of this canonical data are classical; in this paper the canonical form is used as an input from the theory of pairs of skew-symmetric forms under congruence.

5 Singular chain probes

We first recover the singular Kronecker minimal indices.

For $f\ge1$, define

by

and all other entries zero.

Given a pencil $(A,B)$, define

by

Then

Define the chain defect

Equivalently,

because

By Theorem 3.2, the value

is recovered from a Fourier coefficient of an explicit commutator-word distribution.

Lemma 5.1 (Regular blocks have zero chain defect).

If $(A,B)$ is a regular block, then

\[ \delta_f(A,B)=0 \]

for every $f\ge1$.

5.1 Singular Kronecker blocks

For $e\ge0$, define the rectangular pencil

by

Write

The corresponding skew-symmetric singular block is

on

Thus

Lemma 5.2 (Singular chain defect).

For the singular block $\mathcal K_e$,

\[ \boxed{ \delta_f(\mathcal K_e)=\max(0,f-e). } \]

Theorem 5.3 (Recovery of singular minimal indices).

Let $m_e$ be the number of singular Kronecker blocks of minimal index $e$. Then

\[ \boxed{ \delta_f = \sum_{e\ge0}m_e\max(0,f-e). } \]

Consequently,

\[ \boxed{ m_e=\delta_{e+1}-2\delta_e+\delta_{e-1}, } \]

with the convention

\[ \delta_0=\delta_{-1}=0. \]

Thus the singular Kronecker data is recovered from the Fourier-tagged rectangular distributions associated to the matrices $(F_f,G_f)$.

6 Closed-point Artin probes for the regular part

We now recover the regular elementary divisors.

Let $\xi$ be a closed point of $\mathbb A^1_k$, represented by an irreducible monic polynomial

Let

For $e\ge1$, define

and let

be multiplication by $t$.

The regular hyperbolic block associated to $(\xi,e)$ is

It carries the alternating forms

Equivalently,

where

Let $\eta$ be another closed point and define

The closed-point Artin probe evaluates the pencil at $\theta_{\eta,f}$. On

it gives the alternating form

Since $A_{\xi,e}\otimes1$ is nondegenerate,

Theorem 6.1 (Regular closed-point block pairing).

For regular hyperbolic blocks and closed-point Artin probes,

\[ \boxed{ \dim_k\operatorname{rad}\Omega_{\xi,e;\eta,f} 2\deg(\xi)\min(e,f)\mathbf 1_{\xi=\eta}. } \]

The point at infinity is handled by replacing $t$ by a local affine coordinate at infinity. The formula is invariant under $\operatorname{PGL}_2(k)$.

7 Singular contribution to Artin probes

To recover the regular part, we must subtract the known contribution of the singular blocks to the Artin probes. We record this explicitly.

Let

be the rectangular Kronecker pencil

and let

be the corresponding skew-symmetric singular block.

Let $\eta$ be a closed point and let

Evaluate $\mathcal K_e(t)$ at

Lemma 7.1 (Singular Artin contribution).

For every singular Kronecker block $\mathcal K_e$, the radical of the evaluated Artin form

\[ \mathcal K_e(\theta_{\eta,f}) \]

has $k$-dimension

\[ \boxed{ \deg(\eta)f. } \]

In particular, this contribution is independent of the minimal index $e$.

Corollary 7.2 (Total Artin radical formula).

Let the pencil have singular multiplicities $m_e$, and let

\[ M_{\mathrm{sing}}=\sum_e m_e \]

be the total number of singular Kronecker blocks. Let its regular divisor be

\[ \mathcal D_{\mathrm{reg}} = \sum_{\xi,e}m_{\xi,e}[\xi,e]. \]

For the closed-point Artin probe ($\eta,f$), the total radical dimension is

\[ \boxed{ \operatorname{rad}_{\eta,f} M_{\mathrm{sing}}\deg(\eta)f + 2\deg(\eta)\sum_e m_{\eta,e}\min(e,f). } \]

Since the singular multiplicities $m_e$ are recovered from the chain probes, $M_{\mathrm{sing}}$ is known. Therefore define

Then

Taking finite differences gives

and therefore

Thus the regular elementary-divisor multiplicities at every closed point $\eta$ are recovered.

8 Word realization of Artin probes

We explain why the Artin radical dimensions above are observable from commutator-word distributions.

Let

Choose a $k$-basis

of $R$, where

Write multiplication in $R$ as

An $R$-valued variable

is represented by $d$ ordinary $V$-variables

through

The expression

has $d$ $k$-coordinates, each of which is a finite $k$-linear combination of commutators, with coefficients determined by the structure constants of $R$ and the coordinates of $\theta$. Hence the distribution of this expression is an explicit commutator-word distribution.

Let

be a generating additive character. Since $R$ is Frobenius, additive characters are parametrized by $R$-linear functionals. The Fourier coefficient associated to $\chi$ gives

Thus the radical size, and hence its $k$-dimension, is observable.

This applies to the total pencil. By Sections 6 and 7, the observed Artin radical dimensions recover the regular elementary-divisor multiplicities after subtracting the already known singular contribution.

9 Finiteness of the probe family

Fix

For singular probes, it suffices to use

Indeed, a singular block $\mathcal K_e$ has dimension $2e+1$, hence $e\le (N-1)/2$, and the formula

uses $f\le e+1$.

For regular probes, a block $(\eta,e)$ can occur only if

Therefore one only needs closed points $\eta$ with

and exponents

To recover $m_{\eta,e}$ by second differences, use probes up to one additional order:

Since $k$ is finite, there are only finitely many closed points of each bounded degree. Thus, for fixed $k$ and $N$, the total family of probes is finite.

10 Orbit-coding and the intrinsic projective invariant

The previous reconstruction is made after choosing a basis of the target $W=k^2$. A different basis acts by

on the parameter line $\mathbb P^1$. Thus the intrinsic regular datum is not the marked divisor

but its orbit

We now formalize the finite orbit-coding step.

Let

be the finite set of decorated regular divisors

satisfying

Choose an injective code

Let

Define

Because $B_0>|\operatorname{PGL}_2(k)|$, the base-$B_0$ expansion of $\mathcal O_{B_0}(\mathcal D)$ has no carry ambiguity and determines the multiset

Since $c$ is injective, this determines the orbit

The singular multiplicities $m_e$ are invariant under target basis change. Thus the pair

is the intrinsic projective canonical datum recovered from the word distributions.

11 Complete pencil theorem

We now prove the main theorem.

Theorem 11.1 (Complete commutator-word tomography of alternating pencils).

Fix

\[ k=\mathbb F_p,\qquad p\neq2, \]

and

\[ N\ge0. \]

There exists an explicit finite family of Fourier-tagged commutator-word distributions such that, for every alternating pencil

\[ \beta:\Lambda^2_k V\to k^2, \qquad \dim V=N, \]

the values of these distributions determine the full skew-symmetric Kronecker canonical data of $\beta$, namely:

• the multiplicities $m_e$ of all singular Kronecker blocks;

• the $\operatorname{PGL}_2(k)$-orbit of the regular decorated divisor

\[ \mathcal D_{\mathrm{reg}}\sum_{\xi,e}m_{\xi,e}[\xi,e]. \]

Equivalently, these distributions determine the projective congruence class of the alternating pencil.

12 Consequence for class-two exponent-$p$ groups

Corollary 12.1 (Complete word-moment invariant for groups with $\dim G'=2$).

Let $G$ and $H$ be special class-two exponent-$p$ groups with

\[ \dim_{\mathbb F_p}G'=\dim_{\mathbb F_p}H'=2. \]

Let

\[ \beta_G:\Lambda^2(G/Z(G))\to G', \]

\[ \beta_H:\Lambda^2(H/Z(H))\to H' \]

be their commutator tensors. Then $G\cong H$ if and only if the finite family of Fourier-tagged commutator-word distributions of Theorem 11.1 agrees for $G$ and $H$, after the natural projective target orbit-coding.

13 Remarks

13.1 Exact distributions, not sampling

The theorem uses exact finite distributions and Fourier coefficients. It does not address how many random samples are needed to estimate these probabilities.

13.2 Efficiency

The orbit-code is finite but may be large. The result is a structural observability theorem, not an optimized group-isomorphism algorithm.

13.3 Relation to rank-zeta tomography

Scalar-extension rank tomography recovers rank distributions and zeta functions of rank loci. For pencils, rank-zeta data sees the reduced arithmetic support of the degeneracy divisor but does not by itself recover singular minimal indices or elementary-divisor thicknesses. The present theorem adds chain probes and Artin probes to recover the full canonical data.

13.4 Relation to word-map fibre enumeration

The Fourier/rank mechanism is close in spirit to known fibre computations for word maps on class-two exponent-$p$ groups. The new feature here is the design of rectangular and Artin probes that extract the complete Kronecker canonical data of an alternating pencil.