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

Le : 27/01/2015 14h00
Par : Nguyen Dang Ho Hai (IHÉS et Hue, Vietnam)
Lieu : I 001
Lien web :
Résumé : Given a prime p, let Kred(U) denote the Grothendieck group generated by the isomorphism classes of reduced injectives unstable modules over the mod p Steenrod algebra. We explain in this talk a proof of the following conjecture of Lionel Schwartz: "The operator induced by Lannes' T-functor on Kred(U) is diagonalisable over the field of rational numbers and has all powers of p as eigenvalues". The main ingredients of the proof consist of a formula of Harris and Shank for the action of Lannes' T-functor on indecomposable injective unstable modules, a description of the Grothendieck group of the semigroup ring of matrices, and a special case of Stickelberger's theorem in the theory of solving polynomial systems.