Let
S be a finite group with a character, sgn, of order 2, and
S′ its central extension by a group
Z = ⟨
z⟩ of order 2. A representation π of
S′ is called
spin if π(
zσ′) = −π(σ′) (σ′ ∈
S′), and the set of all equivalence classes of spin irreducible representations (= IRs) of
S′ is called the
spin dual of
S′. Take a finite number of such triplets (
Sj′,
zj, sgn
j) (1 ≤
j ≤
m). We define twisted central product
S′ =
S1′ $\hat{*}$
S2′ $\hat{*}$ … $\hat{*}$
Sm′ as a double covering of
S =
S1 × … ×
Sm,
Sj =
Sj′/⟨z
j⟩, and for spin IRs π
j of
Sj′, define twisted central product π = π
1 $\hat{*}$ π
2 $\hat{*}$ … $\hat{*}$ π
m as a spin IR of
S′. We study their characters and prove that the set of spin IRs π of this type gives a complete set of representatives of the spin dual of
S′. These results are applied to the case of representation groups
S′ for
S = 𝔖
n and 𝔄
n, and their (Frobenius-)Young type subgroups.
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