Thursday 5 December 2024 at 2 pm, as part of the Al@Bicocca seminar cycle, Lorenzo Guerra (University of Milan-Bicocca) will talk about
Title: The cohomology of the symmetric groups, Hopf rings and extended symmetric powers
Abstract: The direct sum of the cohomology groups H∗(Σn; V⊗n) of the symmetric group Σn on n objects, with coefficients in the representation given by the nth tensor power of a graded algebra V, is endowed with a very rich algebraic structure. Precisely, it is a Hopf ring, a ring object in the category of cocommutative coalgebras. In my talk, I will recall the definition of Hopf ring, and I will show the aforementioned structure can be used to effectively perform computations in the cohomology rings above. I will then discuss the relation with the cohomology of the extended symmetric powers of topological spaces, which are widely used by topologists. This is mostly based on an article published in 2023 by myself, Paolo Salvatore, and Dev Sinha. Finally, I will briefly discuss applications of this framework (some of them more recent): an algebraic one to finite reflection groups, and a topological and representation theoretic one to unordered flag manifolds.
Information to attend
The seminar will be held in Room 3014 (3rd Floor) - U5 Building, University of Milan-Bicocca and will also be available online at the following Webex link