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THE THICKNESS OF SCHUBERT CELLS AS INCIDENCE STRUCTURES

Part of: Algebraic combinatorics

Published online by Cambridge University Press:  02 October 2019

JOHN BAMBERG Open the ORCID record for JOHN BAMBERG [Opens in a new window] ,
ARUN RAM Open the ORCID record for ARUN RAM [Opens in a new window]  and
JON XU Open the ORCID record for JON XU [Opens in a new window]
JOHN BAMBERG
Affiliation:
The University of Western Australia, Australia email john.bamberg@uwa.edu.au
ARUN RAM
Affiliation:
The University of Melbourne, Australia email aram@unimelb.edu.au
JON XU*
Affiliation:
The University of Melbourne, Australia email jonxu88@gmail.com
*
✉jonxu88@gmail.com
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Abstract

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.

Keywords

ovoidsChevalley groupsSchubert cellsfinite geometryrepresentation theory

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 2 , October 2020 , pp. 145 - 156
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

It is a pleasure to thank all the institutions that have supported our work on this paper, in particular, the University of Melbourne, the University of Western Australia, and the Australian Research Council (grants DP1201001942, DP130100674 and FT120100036).

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