For a prime p, we show that uniqueness of factorization into irreducible \(\Sigma _{p^2}\)-invariant representations of \({\mathbb Z}/p \wr {\mathbb Z}/p\) holds if and only if \(p=2\). We also show nonuniqueness of factorization for \(\Sigma _8\)-invariant representations of \(D_8 \wr {\mathbb Z}/2\). The representation ring of \(\Sigma _{p^2}\)-invariant representations of \({\mathbb Z}/p \wr {\mathbb Z}/p\) is determined completely when p equals two or three.

中文翻译：

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对称群的融合不变表示

对于素数p，我们证明分解为不可约\(\Sigma _{p^2}\)不变表示\({\mathbb Z}/p \wr {\mathbb Z}/p\)的唯一性成立当且仅当\(p=2\)。我们还展示了\(\Sigma _8\)的因式分解的非唯一性- D_8 \wr {\mathbb Z}/2\)的不变表示。当p等于 2 或 3时，\(\Sigma _{p^2}\)的表示环 - {\mathbb Z}/p \wr {\mathbb Z}/p\)的不变表示完全确定。