Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T19:35:16.091Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
ARUN RAM
Affiliation:
The University of Melbourne, Australia email [email protected]
JON XU*
Affiliation:
The University of Melbourne, Australia email [email protected]
*
✉[email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).

References

Ball, S. and Weiner, Z., An introduction to finite geometry, Notes, version September 2011, available from https://mat-web.upc.edu/people/simeon.michael.ball.Google Scholar
Birkhoff, G., Lattice Theory, American Mathematical Society Colloquium Publ., Vol. XXV (American Mathematical Society, Providence, RI, 1948).Google Scholar
Bourbaki, N., Lie Groups and Lie Algebras. Chapters 4–6, Elements of Mathematics (Springer, Berlin, 2002), Translated from the 1968 French original by Andrew Pressley.Google Scholar
Brown, M., (Hyper)ovals and ovoids in projective spaces, Socrates Intensive Course, Finite Geometry and its Applications Ghent, 3–14 April 2000, http://cage.ugent.be/∼fdc/intensivecourse2/brown_2.pdf.Google Scholar
Buekenhout, F. and Cohen, A. M., Diagram Geometry: Related to Classical Groups and Buildings, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, No. 57 (Springer, Berlin, 2013).Google Scholar
Cameron, P. J., Projective and polar spaces, http://www.maths.qmul.ac.uk/∼pjc/pps/, 2000.Google Scholar
Dembowski, P., Finite Geometries (Springer, Berlin, 1968).Google Scholar
Fulton, W. and Harris, J., Representation Theory. A First Course, Graduate Texts in Mathematics, 129 (Springer, New York, 1991).Google Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, 52 (Springer, New York, 1977).Google Scholar
Hodge, W. V. D. and Pedoe, D., Methods of Algebraic Geometry, Volume 1 (Cambridge University Press, Cambridge, 1947).Google Scholar
Parkinson, J. and Ram, A., Alcove walks, buildings, symmetric functions and representations, Preprint, 2008, arXiv:0807.3602.Google Scholar
Parkinson, J., Ram, A. and Schwer, C., ‘Combinatorics in affine flag varieties’, J. Algebra 321 (2009), 3469–3493.Google Scholar
Seshadri, C. S., Introduction to the Theory of Standard Monomials, 2nd edn, Texts and Readings in Mathematics, 46 (Hindustan Book Agency, New Delhi, 2014).Google Scholar
Shult, E., Points and Lines: Characterizing the Classical Geometries (Springer, Berlin, 2011).Google Scholar
Steinberg, R., Lectures on Chevalley Groups, University Lecture Series, 66 (American Mathematical Society, Providence, RI, 2016), Notes prepared by John Faulkner and Robert Wilson. Revised and corrected edition of the 1968 original.Google Scholar
Taylor, D. E., The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, 9 (Heldermann, Berlin, 1992).Google Scholar
Tits, J., Les groupes simples de Suzuki et Ree, Séminaire Bourbaki, Année 1960–1961, Exposés 205–222 (Société Mathématique de France, Paris, 1961) 65–82.Google Scholar
Tits, J., ‘Ovoïdes à translations’, Rend. Mat. Appl., V. Ser. 21 (1962), 37–59.Google Scholar
Tits, J., Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, 386 (Springer, Berlin, 1974).Google Scholar
Tits, J., Oeuvres/Collected Works, Vols 4 (eds. Buekenhout, F., Mühlherr, B. M., Tignol, J.-P. and van Maldeghem, H.) (European Mathematical Society, Zurich, 2013).Google Scholar
Tits, J., Résumés des cours au Collège de France 1973–2000, Documents Mathématiques, 12 (Société Mathématique de France, Paris, 2013).Google Scholar
Veldkamp, F. D., ‘Geometry over rings’, Handbook of Incidence Geometry (North-Holland, Amsterdam, 1995), 1033–1084.Google Scholar