Irreducible Semilinear Representations

• Irreducible semilinear representations are modules over skew group algebras that cannot be non-trivially decomposed, with groups acting via field automorphisms.
• They exhibit striking rigidity in settings such as the infinite symmetric group, enforcing that only one-dimensional irreducibles exist under faithful actions.
• Extensions through invariant subfields, such as cross-ratio fields, provide exceptional nontrivial cases linked to cohomology and analogs of Hilbert’s Theorem 90.

An irreducible semilinear representation is a finite- or infinite-dimensional module over a skew group algebra, constructed from a group $G$ acting by field automorphisms on a field $K$, which cannot be decomposed non-trivially under this structure. These representations generalize the concept of linear representations by incorporating the possibility of $G$ acting nontrivially on the scalars. The paper of irreducible semilinear representations has revealed precise and striking rigidity phenomena, especially for large permutation groups such as the infinite symmetric group $S_\infty$ and automorphism groups of infinite vector spaces over finite fields. The main results characterize these irreducibles, their dimensions, and their connection to cohomology and classical theorems such as Hilbert's Theorem 90.

1. Definition and Structural Foundations

Let $G$ be a topological group acting by field automorphisms on a field $K$, written as $G \rightarrow \mathrm{Aut}(K)$. A $K$-semilinear representation of $G$ is a $K$-vector space $V$ with a group homomorphism $\rho: G \rightarrow \mathrm{Aut}_{\mathbb{Z}}(V)$ such that for all $g \in G$, $\lambda \in K$, $v \in V$,

$\rho(g)(\lambda v) = g(\lambda)\, \rho(g)(v)$

Equivalently, $V$ becomes a left module over the skew group ring $K\langle G \rangle$, generated by $K$ and $G$ with multiplication

$(a [g]) \cdot (b [h]) = a\, g(b)\, [gh]$

The representation is termed smooth if for every $v \in V$, the stabilizer subgroup $\mathrm{Stab}_G(v)$ is open in $G$ (i.e., $V$ is discrete as a $G$-set). A non-degenerate semilinear representation has every $g \in G$ acting invertibly on $V$. The representation is trivial if the canonical map $V^G \otimes_{K^G} K \rightarrow V$ is an isomorphism, i.e., $V$ is a direct sum of copies of $K$ with $G$ acting only via its action on $K$ (Rovinsky, 2014).

2. Cohomological Perspective and Hilbert’s Theorem 90 Analog

For precompact $G$ (every open subgroup has finite index) and $K$ with a faithful smooth $G$-action, a generalization of Hilbert's Theorem 90 holds: \emph{Every smooth $K$-semilinear $G$-representation is trivial} (Rovinsky, 2014). This leads to the identification of the category of smooth $K$-semilinear $G$-representations with vector spaces over the fixed field $k=K^G$. Cohomologically, this yields

$$H^1(G, K^\times) = 0 \qquad \text{and} \qquad H^1(G,GL_r(K))=\{1\}$$

Thus, the only possible (semilinear) $G$-structures on $K^r$ are the trivial ones, and all such modules split into direct sums of $K$ (Rovinsky, 2014).

When $G$ is not precompact (notably for $S_\infty$ and similar groups), smooth semilinear $K$-modules of finite length still decompose into direct sums of $K$, but the proof requires a limiting argument using exhaustion by finite subgroups (Rovinsky, 2014).

3. Classification of Irreducible Smooth Semilinear Representations

Suppose $G$ is a permutation group such as $S_\infty$ acting faithfully on $K=k(Y)$, the field of rational functions in $\{x_y\mid y\in Y\}$, with $Y$ countable infinite and $G$ acting by coordinate permutation. The main classification theorem states:

Theorem: If $V$ is a smooth irreducible non-degenerate $K$-semilinear representation of $G$ of finite length, then

• $V\cong K$ as a $K$-vector space (i.e., $V$ is one-dimensional over $K$);
• $G$ acts trivially on the $K$-basis, so the $G$-action is entirely through $K$'s field automorphisms;
• no finite-dimensional irreducible smooth $K$-semilinear representation of higher dimension exists.

This result excludes the existence of higher-dimensional irreducibles whenever $G$ acts faithfully on $K$ with all open subgroups having infinite index, as in the infinite symmetric group or the automorphism group of an infinite-dimensional vector space over a finite field (Rovinsky, 2014, Rovinsky, 2022).

In cohomological terms, the isomorphism classes of $K$-semilinear $G$-structures on $K^r$ are parameterized by $H^1(G, GL_r(K))$, which vanishes in this setting (Rovinsky, 2014).

4. Cross-Ratio Fields and Beyond: Exceptional Irreducibility

In scenarios where the base field $k$ is extended appropriately, nontrivial irreducible smooth semilinear representations emerge. Specifically, by passing from $k$ to the cross-ratio subfield $K_a \subset k(S)$ generated by invariants of the form

$r(t,u;v,w) = \frac{(t-u)(v-w)}{(t-v)(u-w)}$

for distinct $t,u,v,w \in S$, one obtains a $G=\operatorname{Sym}(S)$–stable field. Over $K_a$, finite-dimensional smooth semilinear representations correspond to rational representations of $\mathrm{PGL}_{2,k}$:

$$V \longmapsto \widetilde{V} = \operatorname{Hom}_{k(x)}\!\bigl(V,\,k(S)\bigr)$$

This yields a precise correspondence (Theorem 5.23 in (Rovinsky, 2022)), allowing the construction of nontrivial higher-dimensional irreducibles mirroring the algebraic representation theory of $\mathrm{PGL}_{2,k}$. In this context, multiplicities, dimensions, and classifying invariants are entirely determined by the structure of $\mathrm{PGL}_2$-modules and the combinatorics of field invariants (Rovinsky, 2022).

5. Semilinear Representation Theory for Finite Galois Extensions

When $L/K$ is a finite Galois extension and $G$ acts on $L$, with $K=L^G$, the structure of finite-dimensional semilinear representations is governed by the linear representation theory of $H=\ker(G\to \operatorname{Gal}(L/K))$. The classification follows from two results (Taylor, 6 Nov 2025):

• Uniqueness of Extension: $V,W$ in $\operatorname{Rep}_L^\ltimes(G)$ are isomorphic if and only if their restrictions to $H$ are isomorphic as $L$-linear $H$-modules. Furthermore, $\operatorname{Hom}$-spaces base-change from $K$ to $L$.
• Classification: Irreducible $L$-semilinear $G$-modules correspond bijectively to Galois orbits of irreducible $L$-linear $H$-modules, with multiplicities described by a "semilinear Schur index" $m_K^L(W)$.

This framework recovers the classical case (trivial $G$-action on $L$), realizes explicit character theory for semilinear representations, and ties arithmetic questions (e.g., solvability of certain equations) to the extension problem for irreducible modules (Taylor, 6 Nov 2025).

6. Category-Theoretic and Structural Properties

The category $Sm_K(G)$ of smooth $K$-semilinear representations is:

• Semisimple if and only if $G$ is precompact (i.e., for $k(S)$, every module splits into direct sums of $k(S)$ itself) (Rovinsky, 2022).
• Locally noetherian: any finitely generated smooth $K(G)$-module is noetherian (Corollary 4.15, (Rovinsky, 2022)).
• Splitting of morphisms: any morphism between finitely generated objects splits locally after restricting to an open subgroup.

In the presence of a cogenerator field such as $k(S)$, every smooth semilinear module embeds into a product of copies of $k(S)$ (Theorem 5.7, (Rovinsky, 2022)). For invariant subfields like $K_a$, the category is no longer semisimple but has a well-controlled Gabriel spectrum and injective hulls formed from permutation modules coinduced from finite stabilizers.

7. Examples and Counterexamples

Finite group $G$ acting on $K$: Classical Hilbert 90 applies. Every irreducible is $1$-dimensional with trivial semilinear structure (Rovinsky, 2014).

$S_\infty$ or automorphism groups with faithful actions: Only $1$-dimensional smooth irreducible $K$-semilinear representations exist; higher-dimensional irreducibles do not arise (Rovinsky, 2014, Rovinsky, 2022).

Cross-ratio fields: By enlarging the base to $K_a$, higher-dimensional irreducibles become possible and correspond to representations of $\mathrm{PGL}_{2,k}$ (Rovinsky, 2022).

Action not faithful/trivial: If $G$ acts trivially on $K$, then permutation modules $K[G/U]$ possess genuine irreducibles of dimension equal to the number of double cosets $U\backslash G / U$, which disappear for faithful actions and infinite index stabilizers (Rovinsky, 2014).

Schur index phenomena: In finite Galois cases, arithmetic properties (Pell equations, presence of roots of unity, etc.) govern when irreducible $H$-modules extend to semilinear $G$-modules, as detected by the semilinear Schur index $m_K^L(W)$ (Taylor, 6 Nov 2025).

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In summary, the theory of irreducible semilinear representations for permutation groups and related settings reveals a spectrum from rigidity—with only $K$ itself as irreducible in highly symmetric faithful situations—to rich structures indexed by Galois orbits and algebraic invariants in the presence of suitable field extensions and invariant subfields. The underlying mechanism is the deep relationship between group cohomology, field invariants, and module-theoretic induction-restriction, unified by both classical and new analogs of Hilbert's Theorem 90 (Rovinsky, 2014, Rovinsky, 2022, Taylor, 6 Nov 2025).