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

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 Knred generated by indecomposable summands of the mod p cohomology of a rank n elementary abelian p  -group is diagonalizable over QQ, with eigenvalues 1,p,…,pn1,p,…,pn and with multiplicities pn−pn−1,pn−1−pn−2,…,p−1,1pn−pn−1,pn−1−pn−2,…,p−1,1, respectively. Using work of Harris and Shank, we first reduce this to an algebraic question involving the Grothendieck ring G0(Mn,p)G0(Mn,p) of modules over the semigroup ring Fp[End(Fp⊕n)], showing that the induced action of T   on Knred corresponds to the multiplication by an explicit element. In the second step, we establish the separability of the algebra C⊗G0(Mn,p)C⊗G0(Mn,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.