Semilinear Character Theory

• Semilinear Character Theory is a framework that generalizes linear representations by allowing group elements to act via twisted field automorphisms.
• It constructs and classifies irreducible semilinear representations over finite Galois extensions using matrix-cocycle formalism and the semilinear Schur index.
• The theory preserves key character properties such as orthogonality and decomposition while extending classical tools to capture richer algebraic structures.

A semilinear representation of a group $G$ over a field $L$ endowed with a $G$-action by automorphisms generalizes the notion of a linear representation by allowing the group elements to act semilinearly: for $g \in G$, the action is only $L$-linear up to the twist by the $G$-action on $L$. The question of how to construct and classify such representations, and to extend the powerful character-theoretic machinery of linear representation theory to this context, is addressed in the semilinear character theory. The rigorous development of this theory, particularly for the situation where $L/K$ is a finite Galois extension and $G \to \Gal(L/K)$ is surjective, yields a complete classification of irreducible semilinear representations and provides character-theoretic tools that generalize classical results and clarify the structure of the semilinear world (Taylor, 6 Nov 2025).

1. Foundations: Semilinear Representations and Matrix-Cocycle Formalism

Let $L/K$ be a finite Galois extension with Galois group $\Gamma = \Gal(L/K)$. Consider a group $G$ equipped with a surjection $\sigma:G \to \Gamma$, giving a $G$-action on $L$ via $\sigma$, so that $K=L^G$. Denote by $H = \ker(\sigma) \subset G$ the subgroup acting trivially on $L$.

A semilinear $G$-representation over $L$ consists of a finite-dimensional $L$-vector space $V$ and a map $\rho: G \to \mathrm{Aut}_K(V)$ such that for $g\in G$, $\rho(g)$ is $\sigma_g$-semilinear: $\rho(g)(\lambda v) = \sigma_g(\lambda)\rho(g)(v)$ for all $\lambda\in L, v\in V$. This category is denoted by $\mathrm{Rep}_L^\rtimes(G)$.

Upon choosing an $L$-basis, the action is described by matrices $A_g \in \mathrm{GL}_n(L)$ satisfying the twisted cocycle condition: $A_{g_1g_2} = A_{g_1} \cdot \sigma_{g_1}(A_{g_2})$. Equivalently, representations correspond to modules over the twisted group algebra $L\rtimes G$ with multiplication $$(\ell_1 g_1)*(\ell_2 g_2)=\ell_1 \sigma_{g_1}(\ell_2)(g_1g_2)$$.

Restriction to $H$ yields a purely linear $L$-representation $V|_H$ of $H$.

2. Classification: Irreducible Semilinear Representations and the Schur Index

Let $\mathrm{Irr}_L^\rtimes(G)$ denote the isomorphism classes of irreducible semilinear $G$-representations over $L$, and $\mathrm{Irr}_L(H)$ those of irreducible linear representations of $H$ over $L$. The quotient group $\Gamma$ acts on $\mathrm{Irr}_L(H)$ via conjugation on scalars. For each irreducible $V\in\mathrm{Irr}_L^\rtimes(G)$, choose an irreducible $H$-submodule $W\subset V|_H$; the stabilizer subgroup $\Gamma_W\subset \Gamma$ is defined, and define $G_W = \sigma^{-1}(\Gamma_W)$.

The principal bijection (Theorem B) asserts:

• $$V|_H \cong \bigoplus_{\gamma \in \Gamma/\Gamma_W} (\gamma^* W)^{\oplus m(V)}$$,
• $$\mathrm{Ind}_H^G(W)\cong V^{\oplus (|\Gamma_W|/m(V))}$$,
• The set $\mathrm{Irr}_L^\rtimes(G)$ bijects with $\mathrm{Irr}_L(H) / \Gamma$ (the set of $\Gamma$-orbits), with $V\mapsto$ the orbit of $W$, and $m(V)$ divides $|\Gamma_W|$.

Here $m(V)$ is the semilinear Schur index, a positive integer measuring the minimal exponent for which the extension becomes split.

3. Character Theory: Construction and Basic Properties

When $G$ is finite and $|G| \in L^\times$, both $\mathrm{Rep}_L^\rtimes(G)$ and $\mathrm{Rep}_L(H)$ are semisimple. For $V\in \mathrm{Rep}_L^\rtimes(G)$, the character is defined as $\chi_V:H \to L$, $\chi_V(h) = \mathrm{Tr}(\rho(h):V\to V)$, i.e., as the trace of the $L$-linear action of $H$ on $V$.

Characters satisfy:

• $\chi_{V\oplus W} = \chi_V + \chi_W$,
• $\chi_{V\otimes W}(h) = \chi_V(h)\chi_W(h)$,
• For $g\in G$, the conjugate representation $g*V$ has character $g*\chi_V$ defined by $(g*\chi)(h) = \sigma_g(\chi(g^{-1} h g))$.

Characters are class functions on $H$:

$$\mathrm{Fun}(H/\!/H,L) = \{ f:H\to L \mid f(g^{-1} h g) = f(h)\}.$$

The character map $V\mapsto\chi_V$ is injective on isomorphism classes, and the irreducible characters correspond bijectively to the $\Gamma$-orbits in $\mathrm{Irr}_L(H)$.

4. Orthogonality and Decomposition: Inner Products and Endomorphism Rings

The natural inner product on the space of class functions $$\langle f, g \rangle = \frac{1}{|H|}\sum_{h\in H} f(h) g(h^{-1})$$ (valued in $L$) satisfies

$$\langle \chi_V, \chi_W\rangle = \dim_L\mathrm{Hom}_{L[H]}(V|_H, W|_H).$$

A fundamental result (Theorem A) provides an isomorphism

$$L\otimes_K \mathrm{Hom}_{L\rtimes G}(V, W) \simeq \mathrm{Hom}_{L[H]}(V|_H, W|_H),$$

implying

$$\dim_K \mathrm{Hom}_{L\rtimes G}(V, W) = \langle \chi_V, \chi_W \rangle = \dim_L \mathrm{Hom}_{L[H]}(V|_H, W|_H).$$

Therefore, irreducible semilinear characters satisfy orthogonality up to their endomorphism rings over $K$:

• $\langle \chi_{V_i}, \chi_{V_j}\rangle = 0$ for $i\neq j$,
• $\langle \chi_{V_i}, \chi_{V_i}\rangle = \dim_K\End_{L\rtimes G}(V_i)$.

The multiplicities and the structure of the endomorphism ring thus generalize the classical orthogonality and Schur index theory.

5. Relation to Classical Linear Character Theory

Specializing to the case where $G$ acts trivially on $L$ ($\sigma_g=\mathrm{id}_L$), one has $H=G$ and the semilinear theory collapses to classical character theory:

• $\mathrm{Rep}_L^\rtimes(G) = \mathrm{Rep}_L(G)$,
• The classification, characters, bijection of orbits, and orthogonality relations all agree with the standard theory,
• The Schur index $m(V)=1$ and all decomposition rules reduce to the known ones for linear representations.

6. Illustrative Examples

Example 6.1 (Cyclic Group $C_4$ acting on a quadratic field):

Let $L=K(\sqrt{d})$ with $\mathrm{char}(K)\neq 2$, $d\notin K^2$; $G=C_4$ acts via $C_4\to \mathrm{Gal}(L/K)\simeq C_2$ by mapping $y^2$ to the nontrivial automorphism, so $H=\langle y^2\rangle\cong C_2$. Irreducible semilinear $G$-representations correspond to irreducible characters $\chi$ of $H$ fixed by $\Gamma$, with Schur index dividing $2$. The semilinear extension of the sign character exists if and only if the negative Pell equation $x^2 - d y^2 = -1$ has a solution in $K$, leading to a unique extension with $m=1$, otherwise $m=2$ and the corresponding irreducible is $2$-dimensional with endomorphism ring the quaternion algebra $(-1, d)_K$.

Example 6.2 (Semilinear $S_3$-representations):

For $G=S_3$, $L/K$ a quadratic Galois extension, $H\cong C_3$. Irreducible $L$-characters of $C_3$ decompose as $\chi_1, \chi_\omega, \chi_{\omega^2}$ ($\omega$ a primitive third root). The response of the orbits and indices depends on whether $\omega\in K$, $\omega\in L\setminus K$, or $\omega\notin L$, providing various scenarios in which $2$-dimensional semilinear representations arise, always with Schur index $1$.

In all cases, the orthogonality relations $\langle\chi_i, \chi_j\rangle = \delta_{ij}$ recover a complete semilinear character table.

7. Structural Theorems, Extensions, and Open Problems

Theorem A establishes the uniqueness of the extension from $H$ to $G$ in the semilinear case: Two semilinear $G$-representations $V, W$ are isomorphic if and only if their restrictions to $H$ are isomorphic as $L[H]$-modules. The $K$-linear Hom-space between $V$ and $W$ is controlled via scalar extension $L\otimes_K$ by the $L$-linear Hom-space between the underlying $H$-modules.

Notably, this framework extends naturally to twisted group algebras and informs the structure of the decomposition matrices, Schur indices, and the mapping of irreducibles under field automorphisms.

Potential open directions include a systematic character theory for infinite-dimensional and infinite group actions (subject to appropriate faithfulness and smoothness conditions), as considered for permutation-type groups in (Rovinsky, 2014). In those infinite settings, rigidity phenomena emerge: all irreducible, smooth, semilinear modules may be forced to be one-dimensional and essentially trivial, with delta-function characters at the identity. The broader challenge remains to characterize indecomposable injectives and to fully develop ring-theoretic aspects of semilinear character theory in infinite and infinite-type cases.

The generalization provided here unifies and clarifies how linear representation theory and classical character theory sit within a richer semilinear framework, extending core results and suggesting new structures for further representation-theoretic paper (Taylor, 6 Nov 2025, Rovinsky, 2014).