Relative Linear Word Observables and Representation Complexity of Semidirect Products

1 Introduction

A finite group may be represented in many ways: by generators and relations, by matrices, by a polycyclic presentation, by a short straight-line description, by a normal coordinate system, or by an extension

Such representations do more than describe the abstract group. They determine which operations are easy, which observables are available, what data can be recovered, and how much information is visible at bounded complexity.

This paper studies a concrete representation system: marked semidirect products

where $Q$ is a fixed finite group and $M$ is a finite-dimensional $kQ$-module. The marking means that the normal abelian subgroup $M$, the quotient $Q$, and the action of $Q$ on $M$ are part of the represented object. This avoids ambiguities caused by automorphisms of $Q$ or by noncanonical choices of complements.

The central question is: how much of the $kQ$-module $M$ is visible through bounded-complexity relative observables in $M\rtimes Q$? The answer is sharp.

First, relative linear word observables are exactly matrix-rank observables. Given

we form the relative observable

where $y$ ranges over the abelian normal subgroup $M$ and $y^{c_q}$ denotes the scalar multiple $c_qy$. If $k=\mathbb F_p$, this scalar operation is an ordinary group power in $M$. If $k=\mathbb F_q$ with $q>p$, it is part of the marked $k$-linear observable language. Since $M$ is abelian,

More generally, a matrix $A\in M_{s\times t}(kQ)$ gives a tuple of relative observables whose evaluation is exactly the linear map

When the input is uniformly distributed over $M^t$, the output is uniformly distributed on $\operatorname{im}A_M$. Therefore the value distribution determines $\operatorname{rank}A_M$.

Second, the full profile is complete. If

for all matrices $A$ over $kQ$, then $M\cong N$ as $kQ$-modules. Hence the full relative matrix word profile classifies marked semidirect products. The proof uses standard tools from model theory of modules: ranks of matrices determine dimensions of pp-definable subgroups and pp-pairs; pp-pairs, equivalently finitely presented functors, detect indecomposable summands; Krull-Schmidt then recovers the module.

Third, finite-budget behavior separates the representation-theoretic regimes. If $kQ$ is semisimple, primitive central idempotents give bounded complete observables. If $Q=C_{p^a}$ in characteristic $p$, iterated difference observables recover the ranks of radical powers and hence the Jordan partition. In wild modular regimes, finite observables cannot classify generic positive-dimensional families, even though the full profile is complete.

2 Marked Semidirect Products and Relative Observables

Let $k$ be a field, $Q$ a finite group, and $M$ a finite-dimensional left $kQ$-module. We write the associated semidirect product as

The multiplication is

We regard $G_M$ as a marked semidirect product: the subgroup $M$, the quotient $Q$, and the action of representatives of $Q$ by conjugation are part of the representation.

Definition 2.1 (Relative linear word observables).

A relative linear word observable in variables $y_1,\ldots,y_t$ ranging over $M$ is an expression built from:

• multiplication in the abelian group $M$;

• conjugation by marked elements of $Q$;

• scalar multiplication $y\mapsto cy$ for $c\in k$.

Equivalently, each output coordinate is a $kQ$-linear combination of the variables $y_1,\ldots,y_t$.

Remark 2.2 (Ordinary group words).

When $k=\mathbb F_p$, the additive group of $M$ is elementary abelian, and scalar multiplication by $c\in\mathbb F_p$ is the ordinary group power $y\mapsto y^c$. In this prime-field case, the relative linear observables above can be realized by ordinary relative group words using conjugation by $Q$.

When $k=\mathbb F_q$ with $q=p^f$ and $f>1$, scalar multiplication by a general $c\in\mathbb F_q$ is not an ordinary group word operation on the underlying abelian group. For this reason the present paper uses the more precise term relative $k$-linear word observable. The ordinary-word interpretation should be read literally only in the prime-field case or after adding the scalar operations to the marked language.

2.1 Unary observables

For

define the unary relative observable

The order of the product is irrelevant because all factors lie in the abelian subgroup $M$.

Lemma 2.3 (Group-algebra evaluation).

For every $v\in M$,

\[ w_a(v)=a\cdot v. \]

2.2 Matrix observables

Let

It defines a $k$-linear map

by

For each row $i$, define

and let $W_A=(w_1,\ldots,w_s)$.

Proposition 2.4 (Matrix observable evaluation).

For every $\mathbf v=(v_1,\ldots,v_t)\in M^t$,

\[ W_A(\mathbf v)=A_M(\mathbf v). \]

3 Value Distributions and Rank Profiles

Assume now that $k=\mathbb F_q$ is finite. Let $M$ be finite-dimensional over $k$ and sample $\mathbf v\in M^t$ uniformly. Since $A_M:M^t\to M^s$ is linear, the random variable $A_M(\mathbf v)$ is uniformly distributed on $\operatorname{im}A_M$. Hence the distribution of $W_A$ determines

In particular,

Definition 3.1 (Matrix rank profile).

Let $A$ be a finite-dimensional $k$-algebra and $M$ a finite-dimensional left $A$-module. The full matrix rank profile of $M$ is

\[ \operatorname{MRP}_A(M) = \left( \operatorname{rank}_k B_M \right)_{B\in M_{s\times t}(A),\ s,t\ge0}. \]

If $A=kQ$ and $G_M=M\rtimes Q$, the full relative matrix word profile is

\[ \operatorname{RelMRWP}(G_M)=\operatorname{MRP}_{kQ}(M). \]

A finite-budget version is obtained by restricting to matrices $B$ whose entries are described by observables of complexity at most $L$.

Theorem 3.2 (Relative observables equal matrix-rank observables).

Let $G_M=M\rtimes Q$ be a marked semidirect product over a finite field $k$. Then relative $k$-linear multi-word value distributions determine exactly the matrix rank profile of the $kQ$-module $M$:

\[ \operatorname{RelMRWP}(G_M)=\operatorname{MRP}_{kQ}(M). \]

4 Completeness of the Full Matrix Rank Profile

Theorem 4.1 (Full matrix rank profile completeness).

Let $A$ be a finite-dimensional $k$-algebra, and let $M,N$ be finite-dimensional left $A$-modules. If

\[ \operatorname{MRP}_A(M)=\operatorname{MRP}_A(N), \]

then $M\cong N$ as $A$-modules. Consequently, for marked semidirect products $G_M=M\rtimes Q$ and $G_N=N\rtimes Q$,

\[ \operatorname{RelMRWP}(G_M)=\operatorname{RelMRWP}(G_N) \]

implies $G_M\cong G_N$ as marked semidirect products.

4.1 From matrix ranks to pp-pairs

A pp-formula for left $A$-modules is a system of homogeneous linear equations with existentially quantified variables. It has the form

where $C,D$ are matrices over $A$. For a module $M$, the solution set $\phi(M)$ is a $k$-subspace of $M^n$.

Let

be the induced linear map, and let $D_M:M^m\to M^r$ be the map induced by $D$. Then

Equivalently,

Therefore the matrix rank profile determines the dimension of $\phi(M)$ for every pp-formula $\phi$.

If $\psi\le\phi$ are pp-formulas, the pp-pair $\phi/\psi$ has value $\phi(M)/\psi(M)$, and

Therefore the matrix rank profile determines the dimensions of all pp-pairs.

4.2 pp-pairs detect indecomposable summands

For a finite-dimensional algebra $A$, the category of finite-dimensional $A$-modules is Krull-Schmidt. Every finite-dimensional module decomposes uniquely, up to isomorphism and permutation, as a finite direct sum of indecomposable modules.

We use the following standard consequence of the Auslander category of finitely presented functors, equivalently the model theory of modules.

Lemma 4.2 (Indecomposable-detecting pp-pairs).

For every finite-dimensional indecomposable $A$-module $U$, there exists a pp-pair

\[ \phi_U/\psi_U \]

such that, for every finite-dimensional $A$-module $M$,

\[ \dim_k(\phi_U(M)/\psi_U(M)) = m_U(M)\cdot \dim_k D_U, \]

where $m_U(M)$ is the multiplicity of $U$ as a direct summand of $M$, and

\[ D_U=\operatorname{End}_A(U)/\operatorname{rad}\operatorname{End}_A(U). \]

Equivalently, the simple finitely presented functor

\[ S_U=\operatorname{Hom}_A(U,-)/\operatorname{rad}(U,-) \]

is represented by a pp-pair, and its value on $M$ detects the number of direct summands isomorphic to $U$.

Remark 4.3.

The completeness theorem is not a claim that a finite list of observables classifies arbitrary modules. Rather, the full infinite profile is complete. Finite-budget profiles form a hierarchy of approximations to isomorphism. In semisimple fixed-quotient cases this hierarchy collapses at bounded budget; in wild cases it typically does not.

5 Semisimple Fixed Quotients

Let $k=\mathbb F_q$ be finite and let $Q$ be a finite group with $\operatorname{char}k\nmid |Q|$. By Maschke's theorem, $kQ$ is semisimple. Let

be the decomposition of the identity into primitive central idempotents of $kQ$. Let $S_1,\ldots,S_s$ be the corresponding simple $kQ$-modules, and set $d_i=\dim_k S_i$. Every finite-dimensional $kQ$-module decomposes as

The operator $e_i$ acts as the projection onto the $S_i$-isotypic component. Hence

Theorem 5.1 (Bounded observable classification for fixed semisimple quotients).

Fix $k=\mathbb F_q$ and a finite group $Q$ with $\operatorname{char}k\nmid |Q|$. There exists a constant $L=L(k,Q)$ such that, for every finite-dimensional $kQ$-module $M$, the marked semidirect product $G_M=M\rtimes Q$ is classified by the relative identity profiles of unary $k$-linear observables of complexity at most $L$.

Remark 5.2.

If one insists on ordinary group words rather than $k$-linear relative observables, the prime-field case $k=\mathbb F_p$ is literal. Over larger finite fields, the theorem should be read in the marked $k$-linear observable language, or after replacing scalar operations by a fixed marked implementation when such an implementation is part of the representation.

6 Cyclic Modular Quotients and Radical Depth

Let $k=\mathbb F_p$ and

A finite-dimensional $kQ$-module is equivalent to an operator $T$ satisfying $T^{p^a}=I$. Set

In characteristic $p$,

so $N$ is nilpotent. Let $G_T=V\rtimes_T C_{p^a}$.

Define the difference observable

Then

Define iterates by

Then

Theorem 6.1 (Modular cyclic radical-depth classification).

The relative identity profiles of

\[ \Delta^{(e)}(y), \qquad 1\le e\le p^a, \]

determine the Jordan partition of $N=T-I$, hence classify the marked semidirect product $V\rtimes_T C_{p^a}$.

Proposition 6.2 (Pure-difference lower bound).

For every $D\ge1$, there exist two nonisomorphic $kC_{p^a}$-modules, for $a$ large enough, that are indistinguishable by the observables

\[ \Delta^{(e)},\qquad 1\le e\le D. \]

Remark 6.3.

The lower bound concerns the pure-difference hierarchy $\Delta^{(e)}$. In a straight-line program model, certain high radical powers can be probed more efficiently. For example,

\[ t^{p^s}yt^{-p^s}y^{-1} \]

evaluates to

\[ (T^{p^s}-I)y=(T-I)^{p^s}y=N^{p^s}y. \]

Thus the observable complexity of probing $N^e$ depends on the operator-polynomial complexity of $(z-1)^e$ modulo $(z-1)^{p^a}$, not simply on $e$.

7 Elementary Abelian Quotients and Rank Varieties

Let

and let $k$ have characteristic $p$. Then

A finite-dimensional $kE_r$-module is a tuple of commuting nilpotent operators

For every $\lambda=(\lambda_1,\ldots,\lambda_r)\in k^r$, define

The Jordan type of $N_\lambda$ is determined by the ranks of $N_\lambda^e$ for $1\le e\le p$.

Theorem 7.1 (Rank-variety observables).

The relative matrix-rank profile of $M\rtimes E_r$ recovers the rank and Jordan-type data of the restrictions of $M$ to all rank-one shifted cyclic subgroups. In particular, it recovers the usual rank-variety and constant-Jordan-type loci associated to $M$.

Remark 7.2.

Rank varieties and constant Jordan type are therefore contained in the relative matrix-rank observable profile. The full matrix word profile is broader: it includes ranks of arbitrary matrices over $kE_r$, not only ranks of powers of one-parameter nilpotent restrictions.

8 Wild Regimes

Let $A$ be a finite-dimensional algebra over an algebraically closed field $k$. Let $X$ be an irreducible algebraic variety parametrizing a family of $A$-modules

of fixed dimension. Assume the family is generically non-isomorphic: there is a dense open subset of $X$ whose distinct generic points represent nonisomorphic modules.

For a fixed matrix

the map $B_{M_x}:M_x^t\to M_x^s$ depends algebraically on $x$. Hence the function

is maximal on a dense open subset of $X$.

Theorem 8.1 (Generic finite-observable blindness).

Let $(M_x)_{x\in X}$ be an irreducible algebraic family of $A$-modules. For every finite set of matrix-rank observables

\[ \mathcal S=\{B_1,\ldots,B_N\}, \]

there exists a dense open subset $U_{\mathcal S}\subseteq X$ such that all ranks

\[ \operatorname{rank}(B_i)_{M_x} \]

are constant on $U_{\mathcal S}$. If the family is generically non-isomorphic and $\dim X>0$, then no finite set $\mathcal S$ can classify the generic members of the family.

Corollary 8.2.

Let $G_x=M_x\rtimes Q$ be a marked semidirect family with $A=kQ$. For any finite family of relative matrix-rank observables, there is a dense open subset of $X$ on which all corresponding observable profiles are constant. Thus finite observables detect rank-degeneration strata, not generic parameters, in positive-dimensional wild families.

9 Fox-Linear Homomorphism Lifting

We include a further application connecting relative observables with homomorphism counts.

Let

be a finitely presented group. Let $G=M\rtimes Q$ be a split marked semidirect product. Fix a homomorphism $\phi:\Gamma\to Q$, determined by $q_i=\phi(x_i)$ satisfying $r_j(q_1,\ldots,q_k)=1$.

A lift of $\phi$ to $G$ is obtained by choosing $a_i\in M$ and mapping

The relations become linear equations in the $a_i$. Let $\partial r_j/\partial x_i\in\mathbb ZF_k$ be the Fox derivatives. Evaluating at $q_i$ gives elements of $\mathbb ZQ$, hence operators on $M$. Define the Fox-Jacobian

by

Theorem 9.1 (Split Fox-Hom lifting formula).

For the split semidirect product $G=M\rtimes Q$,

\[ |\operatorname{Hom}_\phi(\Gamma,G)| = |\ker J_\phi|. \]

Consequently,

\[ |\operatorname{Hom}(\Gamma,G)| = \sum_{\phi\in\operatorname{Hom}(\Gamma,Q)} |\ker J_\phi|. \]

Remark 9.2 (Nonsplit extensions).

For a nonsplit extension with abelian kernel, the same computation gives an affine system

\[ J_\phi(\mathbf a)+B_c(\phi)=0, \]

where $B_c(\phi)$ is a cocycle-dependent obstruction term. Hence

\[ |\operatorname{Hom}_\phi(\Gamma,G)| = |\ker J_\phi|\, \mathbf 1_{-B_c(\phi)\in\operatorname{im}J_\phi}. \]

10 Marked and Unmarked Semidirect Products

All main classification theorems above concern marked semidirect products. This is the natural setting for representation complexity, because the action module $M$ and quotient $Q$ are part of the representation.

For unmarked groups, additional equivalences appear. If $Q=C_m$, an automorphism of $Q$ may send

Thus the action $T$ may be replaced by $T^u$. Accordingly, unmarked profiles can classify at best the orbit of the action under the relevant automorphism group, unless the subgroup $M$ and quotient $Q$ are intrinsically characterized in the abstract group.

In general, unmarked classification requires quotienting module classification by the natural action of $\operatorname{Aut}(Q)$ and by any further automorphisms of the extension structure. We do not pursue this here. The marked theory is the basic representational object.

11 Summary and Open Problems

The results give the following hierarchy for marked semidirect products $G_M=M\rtimes Q$.

• In the semisimple fixed-quotient case, bounded observables classify arbitrary-dimensional semidirect products.

• In the cyclic modular case, observables $\Delta^{(e)}$ recover radical depth and the Jordan partition.

• In elementary abelian and wild modular cases, relative matrix-rank observables include rank-variety and constant-Jordan-type data; the full profile is complete, but finite profiles are generically blind on positive-dimensional wild families.

Problem 11.1.

For a wild algebra $A$ and a finite-budget observable class, estimate the residual fiber size over finite fields, or the geometric residual fiber dimension over algebraically closed fields.

Problem 11.2.

For tame algebras, determine whether finite or logarithmically growing observable budgets recover the parameters of one-parameter families.

Problem 11.3.

For $kC_{p^a}\cong k[\epsilon]/(\epsilon^{p^a})$, determine the straight-line complexity of constructing the operator $\epsilon^e$ as a relative observable.

Problem 11.4.

Determine conditions under which the abelian normal subgroup $M$ and quotient $Q$ are intrinsic in the abstract group $M\rtimes Q$, so that marked classification descends to unmarked classification.

Problem 11.5.

Describe precisely which finite-budget matrix-rank observables correspond to known rank varieties, constant Jordan type, radical type, and socle type invariants for elementary abelian $p$-groups.