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.
Bibliographical noteFunding Information:
The work of Reiner and Dorpalen-Barry was supported by NSF grant DMS-1601961. The team also sincerely thanks the Polymath Jr. REU faculty organizers (Kira Adaricheva, Ben Brubaker, Pat Devlin, Steven Miller, Alexandra Seceleanu, Adam Sheffer, Yunus Zeytuncu) for their vision and hard work in making this bold new opportunity a reality. We also acknowledge the work of participant Huda Ahmed.
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- Hilbert series
- k-Schur function