Seminar on Combinatorics, Lie Theory, and Topology
# The mod $p$ cohomology of complete unordered flag manifolds in $\mathbb{C}^n$ and $\mathbb{R}^n$

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Europe/Rome

Aula Magna (Department of Mathematics)
### Aula Magna

#### Department of Mathematics

Description

Flag manifolds are topological spaces parametrizing nested subspaces in a fixed vector space. On the complete flag manifold of $\mathbb{C}^n$ and $\mathbb{R}^n$ there is a natural action of the symmetric group on $n$ letters. In this talk I will describe the cohomology of the quotient space of this action with coefficients in prime fields of positive characteristic. After recalling the basic definitions and providing some motivation, I will recall some algebraic and combinatorial properties of the cohomology of extended symmetric powers of topological spaces. I will then apply them to the classifying spaces of wreath products and use some spectral sequence argument to determine the desired cohomology. Most of the content of this talk stems from an ongoing collaboration with S. Jana, A. Maiti, and P. Salvatore.