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Let $p$ be prime and let $T^p$ be the $p$-torus and $\mathbb{Z}/p$ the cyclic group of order $p$ generated by $(12\ldots p)$. Consider the semidirect product $T^p\rtimes\mathbb{Z}/p$ with the natural action.

I would like to know if the representation ring of this semidirect product is known. Alternatively, what is known about the cohomology of the classifying space $H^*(B(T^p\rtimes\mathbb{Z}/p);G)$ for $G$ either $\mathbb{Q}$ or $\mathbb{Z}$?

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For any finite dimensional representation, you can restrict to $T^p$ to get a decomposition into a direct sum of one-dimensional characters, which are given by $p$-tuples of integers. If a given character is diagonal (i.e., of the form $(n,\ldots,n)$), then it is a one-dimensional representation of $\mathbb{Z}/p$, i.e., characterized by a $p$-th root of unity. If a character is non-diagonal, then it is contained in the $p$-dimensional representation containing all cyclic permutations of that character, so the cyclic permutation class characterizes the non-diagonal irreducible representations.

Thus, we can describe the finite dimensional representations as finitely supported sheaves on the stack quotient $[\mathbb{Z}^p_{\mathbb{C}}/(\mathbb{Z}/p)]$. Tensor product is given by lifting to the cover and convolving (i.e., the cyclic symmetry on characters of $T^p$ is preserved).

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Thank you. I was also thinking about restriction to $T^p$, perhaps using induction. I'm looking for a description in terms of generators and relations. Maybe some spectral sequence argument? Bockstein? –  Maxim Stykow Feb 24 at 21:19

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