Generic Odd-Pfaffian Reconstruction from Commutator-Word Moments

Proof.

This is the standard bilinear character average. For fixed $u$, the average over $v$ is $1$ precisely when

and $0$ otherwise. Hence the total average is the proportion of $u$ in the radical of $A_\lambda$, namely

Proof.

By Lemma 3.1, MVMT determines the rank of $A_\lambda$ for every

over every finite extension $K/k$. Therefore it determines, for every $K/k$,

Equivalently, it determines the $K$-rational points of the corank-$\ge3$ locus for every finite extension $K$.

Every closed point of a finite-type $k$-scheme is rational over some finite extension of $k$. Thus the MVMT data determine the set of all closed points of

Since $I_\beta$ is geometrically radical and saturated, it is exactly the homogeneous ideal of this reduced projective subscheme. Hence $I_\beta$ is determined.

Proof.

Both matrices give minimal graded free resolutions of the same homogeneous ideal $I_\beta$:

and

Minimal graded free resolutions over $S$ are unique up to graded chain isomorphism. Since

and

are pure sums of equal shifts, every graded automorphism of these middle terms is given by a constant invertible $n\times n$ matrix over $k$. Therefore the middle maps differ by constant invertible transformations:

for some

Proof.

Since both $M$ and $M'$ are alternating,

From

we get

But

Hence

Multiplying on the left by $P^{-1}$ and on the right by $P^{-\mathsf T}$, we obtain

Set

Then

Therefore

Since

this identity is equivalent to

Thus

By hypothesis,

for some

Therefore

Substituting into

gives

The scalar $c$ is absorbed by a scalar change of target coordinates in $W$. Hence the corresponding tensors are pseudo-isometric.

Proof.

Let

be the projective space of alternating forms on $V$.

Let

be the corank-$\ge3$ Pfaffian variety. Since

is odd, $\Sigma_3$ is the first nontrivial degeneracy locus. It has codimension $3$. Its singular locus is

which has codimension $10$.

A tensor

is the same as a projective linear subspace

For a general such $\mathbb P(W^*)$, the intersection

has expected codimension $3$ in $\mathbb P(W^*)$. Since

for

a general $\mathbb P(W^*)$ avoids $\Sigma_5$. Therefore the intersection occurs in the smooth locus of $\Sigma_3$.

By generic transversality, a general $\mathbb P(W^*)$ intersects the smooth locus of $\Sigma_3$ transversely. Hence $Z_\beta$ is smooth over the algebraic closure. In particular, its homogeneous ideal is saturated and geometrically radical.

The ideal of $\Sigma_3$ is generated by the submaximal Pfaffians of the universal odd alternating matrix. The ideal $I_\beta$ is the pullback of this ideal to $\mathbb P(W^*)$. Since the intersection has expected codimension $3$, the Buchsbaum--Eisenbud acyclicity criterion gives exactness of the associated Pfaffian complex. Minimality follows because all entries of $M_\beta$ are linear and have no constant terms, and the maps are homogeneous of positive degree.

The conditions above are open conditions on $\beta$, and the general-position argument shows that the open set is nonempty.

Proof.

For fixed $\beta=(A_1,\ldots,A_m)$, the equations

are linear equations in the entries of $R$, with coefficients depending linearly on the entries of the forms $A_i$. Thus

is the kernel of a matrix whose entries are polynomial functions of $\beta$. Kernel dimension is upper semicontinuous. Since scalar matrices always belong to $\operatorname{Adj}^+(\beta)$, the condition that the dimension be exactly $1$ is open.

Proof.

Write an endomorphism

in block form relative to

where

Let

viewed as $r\times(r+1)$ matrices. The forms $A_0,A_1$ have block matrices

The condition

is equivalent to the block equations

First consider (1). From $j=0$, since

the first $r$ rows of $P$ are

From $j=1$, since

the last $r$ rows of $P$ are

Comparing the overlapping rows of $P$ gives

for

and the boundary equations give

Thus all off-diagonal entries of $S$ vanish and all diagonal entries are equal. Hence

The same equations then force

Now consider $Q=(q_{ij})$, with

Equation (2) for $j=0$ gives

Equation (2) for $j=1$ gives

From (4), $q_{ii}=0$. From (5), $q_{i+1,i}=0$. If $a<b$, then applying (4) and then (5) gives

This reduces (b-a) by (2), eventually reaching a diagonal or subdiagonal entry, both of which are zero. Hence $q_{a,b}=0$ for $a<b$. The case $a>b$ is similar, using (5) and (4) to reduce to a diagonal or subdiagonal zero. Therefore

The equations for $C$ are the transpose-dual of the equations for $Q$, and the same argument gives

Thus

Therefore

Proof.

By Lemma 7.1 it is open. By Lemma 7.2, the pair $(A_0,A_1)$ has scalar self-adjoint algebra. Adding more alternating forms can only decrease the common self-adjoint algebra. Hence the condition is nonempty for all $m\ge2$.

Proof.

By Lemma 3.2, the MVMT data determine the Pfaffian ideal

By Lemma 4.1, this ideal determines the universal alternating matrix

up to left-right equivalence. By Lemma 5.1 and the condition

this left-right equivalence upgrades to congruence up to target scalar, hence to pseudo-isometry of the alternating tensor.

Proof.

For special class-two exponent-$p$ groups, group isomorphism is equivalent to pseudo-isometry of the commutator tensor. Theorem 8.1 recovers the pseudo-isometry class of $\beta$.

Proof.

The complement of a nonempty Zariski open subset in affine space is contained in a proper algebraic subvariety of bounded degree after spreading out over a finitely generated base. The number of $\mathbb F_q$-points of this complement is

where

Dividing by $q^D$ gives