Filtering cohomology of ordinary and Lagrangian Grassmannians

TL;DR

This paper explores the structure of subalgebras in the cohomology rings of complex Grassmannians and Lagrangian Grassmannians, proposing conjectures and bases related to their Hilbert series and algebraic operations.

Contribution

It introduces new conjectures and bases for understanding the cohomology subalgebras of Grassmannians, extending previous conjectures to Lagrangian Grassmannians.

Findings

Reinterprets the Hilbert series conjecture using k-conjugation

Suggests two conjectural bases for the subalgebras

Proposes an analogous conjecture for Lagrangian Grassmannians

Abstract

This paper studies, for a positive integer $m$, the subalgebra of the cohomology ring of the complex Grassmannians generated by the elements of degree at most $m$. We build in two ways upon a conjecture for the Hilbert series of this subalgebra due to Reiner and Tudose. The first reinterprets it in terms of the operation of $k$-conjugation, suggesting two conjectural bases for the subalgebras that would imply their conjecture. The second introduces an analogous conjecture for the cohomology of Lagrangian Grassmannians.