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QUANTUM ANALOGUES OF SCHUBERT VARIETIES IN THE GRASSMANNIAN

Published online by Cambridge University Press:  01 January 2008

T.H. LENAGAN  and
L. RIGAL
T.H. LENAGAN
Affiliation:
Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland, UK e-mail: tom@maths.ed.ac.uk
L. RIGAL
Affiliation:
Université Jean-Monnet (Saint-Étienne), Faculté des Sciences et Techniques, Département de Mathématiques, 23 rue du Docteur Paul Michelon, 42023 Saint-Étienne Cédex 2, France e-mail: Laurent.Rigal@univ-st-etienne.fr
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Abstract

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We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

Keywords

16W3516P4016S3817B3720G42
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008 , pp. 55 - 70
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

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