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 $L$0. Consider a group $L$1 equipped with a surjection $L$2, giving a $L$3-action on $L$4 via $L$5, so that $L$6. Denote by $L$7 the subgroup acting trivially on $L$8.

A semilinear $L$9-representation over $G$0 consists of a finite-dimensional $G$1-vector space $G$2 and a map $G$3 such that for $G$4, $G$5 is $G$6-semilinear: $G$7 for all $G$8. This category is denoted by $G$9.

Upon choosing an $g \in G$0-basis, the action is described by matrices $g \in G$1 satisfying the twisted cocycle condition: $g \in G$2. Equivalently, representations correspond to modules over the twisted group algebra $g \in G$3 with multiplication $g \in G$4.

Restriction to $g \in G$5 yields a purely linear $g \in G$6-representation $g \in G$7 of $g \in G$8.

2. Classification: Irreducible Semilinear Representations and the Schur Index

Let $g \in G$9 denote the isomorphism classes of irreducible semilinear $L$0-representations over $L$1, and $L$2 those of irreducible linear representations of $L$3 over $L$4. The quotient group $L$5 acts on $L$6 via conjugation on scalars. For each irreducible $L$7, choose an irreducible $L$8-submodule $L$9; the stabilizer subgroup $G$0 is defined, and define $G$1.

The principal bijection (Theorem B) asserts:

• $G$2,
• $G$3,
• The set $G$4 bijects with $G$5 (the set of $G$6-orbits), with $G$7 the orbit of $G$8, and $G$9 divides $L$0.

Here $L$1 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 $L$2 is finite and $L$3, both $L$4 and $L$5 are semisimple. For $L$6, the character is defined as $L$7, $L$8, i.e., as the trace of the $L$9-linear action of $L/K$0 on $L/K$1.

Characters satisfy:

• $L/K$2,
• $L/K$3,
• For $L/K$4, the conjugate representation $L/K$5 has character $L/K$6 defined by $L/K$7.

Characters are class functions on $L/K$8:

$L/K$9

The character map $G \to \Gal(L/K)$0 is injective on isomorphism classes, and the irreducible characters correspond bijectively to the $G \to \Gal(L/K)$1-orbits in $G \to \Gal(L/K)$2.

4. Orthogonality and Decomposition: Inner Products and Endomorphism Rings

The natural inner product on the space of class functions $G \to \Gal(L/K)$3 (valued in $G \to \Gal(L/K)$4) satisfies

$G \to \Gal(L/K)$5

A fundamental result (Theorem A) provides an isomorphism

$G \to \Gal(L/K)$6

implying

$G \to \Gal(L/K)$7

Therefore, irreducible semilinear characters satisfy orthogonality up to their endomorphism rings over $G \to \Gal(L/K)$8:

• $G \to \Gal(L/K)$9 for $L/K$0,
• $L/K$1.

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 $L/K$2 acts trivially on $L/K$3 ($L/K$4), one has $L/K$5 and the semilinear theory collapses to classical character theory:

• $L/K$6,
• The classification, characters, bijection of orbits, and orthogonality relations all agree with the standard theory,
• The Schur index $L/K$7 and all decomposition rules reduce to the known ones for linear representations.

6. Illustrative Examples

Example 6.1 (Cyclic Group $L/K$8 acting on a quadratic field):

Let $L/K$9 with $L$00, $L$01; $L$02 acts via $L$03 by mapping $L$04 to the nontrivial automorphism, so $L$05. Irreducible semilinear $L$06-representations correspond to irreducible characters $L$07 of $L$08 fixed by $L$09, with Schur index dividing $L$10. The semilinear extension of the sign character exists if and only if the negative Pell equation $L$11 has a solution in $L$12, leading to a unique extension with $L$13, otherwise $L$14 and the corresponding irreducible is $L$15-dimensional with endomorphism ring the quaternion algebra $L$16.

Example 6.2 (Semilinear $L$17-representations):

For $L$18, $L$19 a quadratic Galois extension, $L$20. Irreducible $L$21-characters of $L$22 decompose as $L$23 ($L$24 a primitive third root). The response of the orbits and indices depends on whether $L$25, $L$26, or $L$27, providing various scenarios in which $L$28-dimensional semilinear representations arise, always with Schur index $L$29.

In all cases, the orthogonality relations $L$30 recover a complete semilinear character table.

7. Structural Theorems, Extensions, and Open Problems

Theorem A establishes the uniqueness of the extension from $L$31 to $L$32 in the semilinear case: Two semilinear $L$33-representations $L$34 are isomorphic if and only if their restrictions to $L$35 are isomorphic as $L$36-modules. The $L$37-linear Hom-space between $L$38 and $L$39 is controlled via scalar extension $L$40 by the $L$41-linear Hom-space between the underlying $L$42-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 study (Taylor, 6 Nov 2025, Rovinsky, 2014).