#### Vol. 15, No. 2, 2022

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Filtering cohomology of ordinary and Lagrangian Grassmannians

### Victor Reiner and Galen Dorpalen-Barry

Vol. 15 (2022), No. 2, 271–288
##### Abstract

We study, 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.

##### Keywords
Grassmannian, Lagrangian, Hilbert series, $q$-binomial, $k$-conjugation, $k$-Schur function
##### Mathematical Subject Classification
Primary: 05E05, 05E14, 14N15