LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-27

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Rephasing changes factor sets by coboundaries

Statement

For $c : Q \to C^{\times}$ with $c (1) = 1$ , $P_{c} (q) = c (q) P (q)$ has factor set $α_{c} (q, r) = c (q) c (r) c (q r)^{- 1} α (q, r)$ . Hence rephasing preserves the cohomology class.

Facts & Assumptions

Given: A finite group $Q$ , a normalized projective representation $P : Q \to GL (V)$ with factor set $α$ , and a function $c : Q \to C^{\times}$ with $c (1) = 1$ .

[F1]

$P (1) = {id}_{V}$ and $P (q) P (r) = α (q, r) P (q r)$ for all $q, r \in Q$ , with $α (q, r) \in C^{\times}$ . (Projective representations and normalized factor sets).

[F2]

$α (1, q) = α (q, 1) = 1$ for all $q \in Q$ . (The factor set satisfies the two-cocycle equation).

[F3]

For an abelian group $M$ with $G$ -action written additively, the two-coboundary of a normalized one-cochain $u$ is $(δ u) (g, h) = g \cdot u (h) - u (g h) + u (g)$ . (Normalized two-cocycle and two-coboundary).

[F4]

The factor-set model of the second cohomology group is $H^{2} (G, M) = Z^{2} (G, M) / B^{2} (G, M)$ , and replacing a normalized two-cocycle $f$ by $f + δ u$ does not change its class. (Second cohomology by factor sets).

[A1]

Scalar multiples of a linear map compose by multiplying scalars, and scalars commute with composition in $GL (V)$ .

Proof

technique · direct

1.1

$P_{c} (1) = c (1) P (1) = {id}_{V}$ , so $P_{c}$ is normalized.

F1givenalgebra

1.2

Using [F1] and [A1], $P_{c} (q) P_{c} (r) = c (q) c (r) P (q) P (r) = c (q) c (r) α (q, r) P (q r) = c (q) c (r) c (q r)^{- 1} α (q, r) P_{c} (q r)$ .

F1A1algebra

2.1

Step 1.2 exhibits $α_{c} (q, r) = c (q) c (r) c (q r)^{- 1} α (q, r)$ as the factor set of $P_{c}$ ; it takes values in $C^{\times}$ , and it is normalized because $α_{c} (1, q) = c (1) c (q) c (q)^{- 1} α (1, q) = 1$ and $α_{c} (q, 1) = c (q) c (1) c (q)^{- 1} α (q, 1) = 1$ by [F2].

F2step 1.2algebra

3.1

Read in multiplicative notation for the trivial action, the published coboundary of [F3] is $δ c (g, h) = c (g) c (h) c (g h)^{- 1}$ ; step 2.1 therefore says $α_{c} = (δ c) \cdot α$ , the factor set of $P$ multiplied pointwise by the coboundary of $c$ .

F3step 2.1

4.1

By [F4], multiplying a normalized two-cocycle by a coboundary does not change its class in $H^{2} (Q, C^{\times})$ , so $[α_{c}] = [α]$ : rephasing preserves the cohomology class. Conversely, if $α^{'} = (δ c) \cdot α$ for a normalized $c$ , then steps 1.2 and 2.1 show that $α^{'}$ is exactly the factor set of the rephased family $P_{c}$ ; so every factor set cohomologous to $α$ is realized by a normalized rephasing of $P$ .

F4step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

5 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

Britta Späth, Reduction theorems for some global-local conjectures — Remark 1.5(a), printed p. 3 (standard reference, not scraped)
Clara Loh, Group Cohomology, SS 2019 — coboundaries and H^2 (standard reference, not scraped)