A proof of Schwartz's conjecture about the eigenvalues of Lannes' T-functor

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• 作者：Nguyen Dang Ho Hai ^{nguyendanghohai@husc.edu.vn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Unstable modules ; Lannes' T-functor ; Modular representation
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：445
• 期：Complete
• 页码：115-124
• 全文大小：313 K

文摘

This paper gives an algebraic proof of a conjecture due to Lionel Schwartz which asserts that the linear operator induced by Lannes' T  -functor on the Grothendieck group $K_n^{\mathit{red}}$ generated by indecomposable summands of the mod p cohomology of a rank n elementary abelian p  -group is diagonalizable over Q$\mathbb{Q}$, with eigenvalues 1,p,…,pⁿ$1,p,\dots ,p^n$ and with multiplicities pⁿ−pⁿ⁻¹,pⁿ⁻¹−pⁿ⁻²,…,p−1,1$p^n-p^{n-1},p^{n-1}-p^{n-2},\dots ,p-1,1$, respectively. Using work of Harris and Shank, we first reduce this to an algebraic question involving the Grothendieck ring G₀(M_n,p)${\mathrm{G}}_0({\mathrm{M}}_{n,p})$ of modules over the semigroup ring ${\mathbb{F}}_p[\mathrm{End}({\mathbb{F}}_p^{\oplus n})]$, showing that the induced action of T   on $K_n^{\mathit{red}}$ corresponds to the multiplication by an explicit element. In the second step, we establish the separability of the algebra C⊗G₀(M_n,p)$\mathbb{C}\otimes {\mathrm{G}}_0(M_{n,p})$, from which the diagonalizability and the computation of the eigenvalues and their multiplicities follow easily. The arguments use ingredients from the theory of Brauer characters of finite groups.