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Schur index as minimal realization multiplicity

Statement

Assume the Axiom of Choice. Let $χ$ be an irreducible complex character of a finite group and put $K = Q (χ)$ . Its Schur index $m_{K} (χ)$ is the least positive integer $r$ for which the character $r χ$ is afforded by a $K$ -representation. Consequently $χ$ itself is realizable over $K$ if and only if $m_{K} (χ) = 1$ .

Facts & Assumptions

Given: AC, an irreducible complex character $χ$ , and $K = Q (χ)$ .

[L1]

The irreducible $K$ -representation in the Schur-index definition has scalar extension $m_{K} (χ) U$ , where $U$ affords $χ$ (The Schur index of an irreducible character, Scalar extension of an irreducible finite-group representation).

[L2]

A field of definition means a $K$ -model whose complex scalar extension is equivalent to the given representation (Character fields and fields of definition).

[L3]

Under ZFC, simple $K [G]$ -modules correspond bijectively to Galois orbits of simple modules after extension to the chosen finite splitting field. Thus $U$ occurs after scalar extension of exactly one irreducible $K$ -module, namely the module $V$ in [L1] (Galois orbits classify simple modules after splitting base change).

[L4]

Every finite-dimensional $K$ -representation of $G$ is completely reducible (If $char k ∤ ∣ G ∣$ , every finite-dimensional representation of $G$ is completely reducible).

[L5]

Finite-dimensional complex representations of $G$ are determined by their characters (Finite-dimensional complex representations of a finite group are determined up to isomorphism by their characters).

Proof

technique · direct

1.1L1given

Under the stated AC premise, let $V$ be the irreducible $K$ -representation from the Schur-index definition. By [L1], its scalar extension has character $m_{K} (χ) χ$ , so $m_{K} (χ) χ$ is afforded over $K$ .

1.2L1L3L4L5givenalgebra

Conversely, if $r χ$ is afforded by a $K$ -representation $W$ , [L4] decomposes $W$ into irreducible $K$ -summands, while [L5] identifies its complex scalar extension with $U^{\oplus r}$ . By [L3], every summand that contributes $U$ is isomorphic to $V$ , and by [L1] each copy contributes $U$ with multiplicity $m_{K} (χ)$ . Hence $m_{K} (χ) ∣ r$ .

2.1L2step 1.1step 1.2∎

Step 1.1 attains $r = m_{K} (χ)$ and step 1.2 excludes every smaller positive $r$ , so this is the least such multiplicity. With $r = 1$ , [L2] gives exactly the stated realizability criterion.

Depends on

Used by

The trivial character has Schur index one Example

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gabor Wiese, Galois Representations, Definition 2.5.12 through Remark 2.5.15 (standard reference, not scraped)
Weizhe Zheng, Lectures on Algebra, Corollary 4.3.4 (standard reference, not scraped)