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arXiv · 2011.03179

Filtering cohomology of ordinary and Lagrangian Grassmannians

The 2020 Polymath Jr. REU "q-binomials·the Grassmannian group"·:·Huda Ahmed·Rasiel Chishti·Yu-Cheng Chiu·Galen Dorpalen-Barry·Jeremy Ellis·David Fang·Michael Feigen·Jonathan Feigert·Mabel González·Dylan Harker·Jiaye Wei·Bhavna Joshi·Gandhar Kulkarni·Kapil Lad·Zhen Liu·Ma Mingyang·Lance Myers·Arjun Nigam·Tudor Popescu·Victor Reiner·Zijian Rong·Eunice Sukarto·Leonardo Mendez Villamil·Chuanyi Wang·Napoleon Wang·Ajmain Yamin·Jeffery Yu·Matthew Yu·Yuanning Zhang·Ziye Zhu·Chen Zijian
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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.

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The 2020 Polymath Jr. REU "q-binomials, the Grassmannian group", :, Huda Ahmed, Rasiel Chishti, Yu-Cheng Chiu, Galen Dorpalen-Barry, Jeremy Ellis, David Fang, Michael Feigen, Jonathan Feigert, Mabel González, Dylan Harker, Jiaye Wei, Bhavna Joshi, Gandhar Kulkarni, Kapil Lad, Zhen Liu, Ma Mingyang, Lance Myers, Arjun Nigam, Tudor Popescu, Victor Reiner, Zijian Rong, Eunice Sukarto, Leonardo Mendez Villamil, Chuanyi Wang, Napoleon Wang, Ajmain Yamin, Jeffery Yu, Matthew Yu, Yuanning Zhang, Ziye Zhu, Chen Zijian. 2020-11-06. Filtering cohomology of ordinary and Lagrangian Grassmannians. https://doi.org/10.2140/involve.2022.15.271

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