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An invariant irreducible representation extends to its inertia group exactly when the Clifford obstruction vanishes

Statement

Let $N ⊴ G$ be finite groups and let $ρ : N \to GL (S)$ be an irreducible representation on a nonzero finite-dimensional complex space $S$ , with inertia group $I = I_{G} (θ)$ for its character $θ$ . Then $ρ$ extends to a representation $\tilde{ρ} : I \to GL (S)$ with $\tilde{ρ} ∣_{N} = ρ$ if and only if the Clifford obstruction $[α] \in H^{2} (I / N, C^{\times})$ of The Clifford obstruction class of an invariant irreducible representation is zero. In the degenerate case $I = N$ both conditions hold automatically.

Facts & Assumptions

Given: Finite groups $N ⊴ G$ , an irreducible finite-dimensional complex representation $ρ : N \to GL (S)$ with $S \neq 0$ , its character $θ$ , the inertia group $I = I_{G} (θ)$ , the quotient $Q = I / N$ , and projective inertia operators $P$ with normalized two-cocycle $α$ on $Q$ as provided by An invariant irreducible normal representation yields projective inertia operators.

[F1]

There are $P (i) \in GL (S)$ with $P (1) = {id}_{S}$ , $P (n) = ρ (n)$ , $P (n i) = ρ (n) P (i)$ , $P (i n) = P (i) ρ (n)$ and $P (i) P (j) = α (i N, j N) P (i j)$ for a normalized two-cocycle $α$ on $Q = I / N$ ; every second family with the same normalization and identities is $P^{'} (i) = c (i N) P (i)$ for a cochain $c : Q \to C^{\times}$ with $c (N) = 1$ , and then the factor set of $P^{'}$ is $c (q) c (r) c (q r)^{- 1} α (q, r)$ . (An invariant irreducible normal representation yields projective inertia operators).

[F2]

The Clifford obstruction $[α] \in H^{2} (Q, C^{\times})$ is the class of the factor set of any such family, and it does not depend on the family. (The Clifford obstruction class of an invariant irreducible representation).

[F3]

$H^{2} (G, M) = Z^{2} (G, M) / B^{2} (G, M)$ , and replacing a normalized two-cocycle $f$ by $f + δ u$ does not change its class; in particular a class is zero exactly when the cocycle is a coboundary. (Second cohomology by factor sets).

[F4]

With $C^{\times}$ written multiplicatively and trivial action, the two-coboundary of a normalized one-cochain $c$ is $δ c (g, h) = c (g) c (h) c (g h)^{- 1}$ , and $c$ normalized means $c (1) = 1$ . (Normalized two-cocycle and two-coboundary).

[F5]

An extension of $ρ$ to $I$ is a representation $\tilde{ρ} : I \to GL (S)$ with $\tilde{ρ} ∣_{N} = ρ$ . (An extension of a normal subgroup representation).

[F6]

Replacing a normalized projective representation by a rephasing $P_{c} (q) = c (q) P (q)$ with $c (1) = 1$ changes its factor set to $c (q) c (r) c (q r)^{- 1} α (q, r)$ . (Rephasing changes factor sets by coboundaries).

Proof

technique · direct

1.1

Suppose first that $ρ$ extends to a representation $\tilde{ρ} : I \to GL (S)$ , so that $\tilde{ρ} ∣_{N} = ρ$ by [F5], and set $P (i) : = \tilde{ρ} (i)$ , $i \in I$ . Then $P (n) = ρ (n)$ , $P (n i) = ρ (n) P (i)$ and $P (i n) = P (i) ρ (n)$ because $\tilde{ρ}$ is a homomorphism, and $P (i) P (j) = P (i j) = 1 \cdot P (i j)$ for all $i, j$ , so $P$ is a normalized family whose factor set is the constant function $1$ on $Q \times Q$ . By [F2] the Clifford obstruction equals the class of this constant cocycle, so $[α] = [1] = 0$ in $H^{2} (Q, C^{\times})$ .

F1F2F5given

1.2

Suppose now that $[α] = 0$ in $H^{2} (Q, C^{\times})$ , where $α$ is the factor set of the family $P$ of [F1]. Since $H^{2} = Z^{2} / B^{2}$ by [F3] and $Z^{2}$ is a group under pointwise multiplication with identity the constant cocycle $1$ , the triviality of the class of $α$ says that $α$ lies in $B^{2}$ , that is, there is a normalized one-cochain $c : Q \to C^{\times}$ with $α (q, r) = c (q) c (r) c (q r)^{- 1}$ for all $q, r \in Q$ , which is exactly $α = δ c$ in the notation of [F4].

F3F4givenchoose

2.1

Define $P^{'} (i) : = c (i N)^{- 1} P (i)$ for $i \in I$ . Then $P^{'} (1) = c (N)^{- 1} {id}_{S} = {id}_{S}$ and $P^{'} (n) = c (N)^{- 1} ρ (n) = ρ (n)$ for $n \in N$ , since $c$ is normalized by step 1.2. By [F6] the rephasing $P^{'}$ of $P$ by the function $i \mapsto c (i N)^{- 1}$ has factor set $c (i N)^{- 1} c (j N)^{- 1} c (i j N) α (i N, j N)$ , which equals $1$ for all $i, j \in I$ by the coboundary relation of step 1.2. Hence $P^{'} (i) P^{'} (j) = P^{'} (i j)$ for all $i, j$ , so $i \mapsto P^{'} (i)$ is a group homomorphism $I \to GL (S)$ , that is, a representation of $I$ restricting to $ρ$ on $N$ : an extension of $ρ$ in the sense of [F5].

F1F5F6step 1.1step 1.2algebra

3.1

Steps 1.1 and 2.1 prove the two implications for an arbitrary invariant irreducible $ρ$ : extendibility forces $[α] = 0$ , and $[α] = 0$ produces an extension. If $I = N$ , then $Q$ is the trivial group, $P = ρ$ is a normalized family with factor set $1$ by [F1], so $[α] = 0$ by [F2], and the identity map $ρ : N \to GL (S)$ is a representation of $I = N$ restricting to $ρ$ , so both conditions hold automatically; this is the degenerate case of the statement, and no separate construction is needed.

F1F2F5step 1.1step 2.1∎

Depends on

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Dependency tree · two levels

16 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 — Lemma 1.8(d) and the discussion after Remark 1.9, printed pp. 3–4 (standard reference, not scraped)
Tammo tom Dieck, Representation Theory — Remarks (4.2.5)–(4.2.6), printed p. 57 (standard reference, not scraped)