On a conjecture of Lionel Schwartz about the eigenvalues of Lannes’ T-functor. (À propos d’une conjecture de Lionel Schwartz sur les valeurs propres du foncteur T de Lannes.) (English. French summary) Zbl 1318.55015
Let \(p\) be a prime, \(A\) be the mod \(p\) Steenrod algebra, \(\mathcal{U}\) be the category of unstable \(A\)-modules and \(V_{n}=(\mathbb{Z}/p\mathbb{Z})^{n}\). We denote by \(K^{red}_{n}(\mathcal{U})\) the Grothendieck group generated by the isomorphism classes of indecomposable \(A\)-module summands of \(H^{*}(BV_{n})\).
The aim of this paper is to prove (Lionel Schwartz’ conjecture) that the operator induced by Lannes’ T-functor on the vector space \(\mathbb{Q}\otimes_{\mathbb{Z}}K^{red}_{n}(\mathcal{U})\) is diagonalisable with eigenvalues \(1, p, ..., p^{n-1}, p^{n}\) of multiplicities \(p^{n}-p^{n-1}, p^{n-1}-p^{n-2},..., p-1, 1\) respectively.
The aim of this paper is to prove (Lionel Schwartz’ conjecture) that the operator induced by Lannes’ T-functor on the vector space \(\mathbb{Q}\otimes_{\mathbb{Z}}K^{red}_{n}(\mathcal{U})\) is diagonalisable with eigenvalues \(1, p, ..., p^{n-1}, p^{n}\) of multiplicities \(p^{n}-p^{n-1}, p^{n-1}-p^{n-2},..., p-1, 1\) respectively.
Reviewer: Saïd Zarati (Tunis)
MSC:
| 55S10 | Steenrod algebra |
Keywords:
Steenrod algebra; Unstable modules over the Steenrod algebra; Reduced injectives; Lannes’ T-functorReferences:
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