Clifford algebras, symmetric spaces and cohomology rings of Grassmannians

TL;DR

This paper explores the cohomology rings of various Grassmannians and Lagrangian Grassmannians over different fields, providing explicit generators, relations, and connections to Clifford algebras and symmetric spaces.

Contribution

It offers explicit descriptions of the cohomology rings of Grassmannians as symmetric spaces and introduces filtered deformations related to Clifford algebras, enhancing conceptual understanding.

Findings

Explicit generators and relations for cohomology rings

Identification of filtered deformations linked to Clifford algebras

Unified framework for Grassmannians as symmetric spaces

Abstract

We study various kinds of Grassmannians or Lagrangian Grassmannians over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, all of which can be expressed as $\mathbb{G}/\mathbb{P}$ where $\mathbb{G}$ is a classical group and $\mathbb{P}$ is a parabolic subgroup of $\mathbb{G}$ with abelian unipotent radical. The same Grassmannians can also be realized as (classical) compact symmetric spaces $G/K$. We give explicit generators and relations for the de Rham cohomology rings of $\mathbb{G}/\mathbb{P}\cong G/K$. At the same time we describe certain filtered deformations of these rings, related to Clifford algebras and spin modules. While the cohomology rings are of our primary interest, the filtered setting of $K$-invariants in the Clifford algebra actually provides a more conceptual framework for the results we obtain.