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
Milestones
Received: 21 March 2021
Revised: 10 September 2021
Accepted: 11 September 2021
Published: 29 July 2022

Communicated by Jim Haglund
Authors
Victor Reiner
School of Mathematics
University of Minnesota
Minneapolis, MN
United States
Galen Dorpalen-Barry
Fakultät für Mathematik
Ruhr-Universität Bochum
Bochum
Germany