Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T07:07:56.905Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

1.Bruns, W. and Vetter, U., Determinantal rings, Lecture Notes in Mathematics, No. 1327 (Springer-Verlag, 1988).CrossRefGoogle Scholar
2.Jørgensen, P. and Zhang, J.J., Gourmet's guide to Gorensteinness, Adv. in Math. 151 (2000), no. 2, 313–345.CrossRefGoogle Scholar
3.Gonciulea, N. and Lakshmibai, V., Flag Varieties, Actualités Mathématiques (Hermann, Paris, 2001).Google Scholar
4.Huang, R. Q. and Zhang, J. J., Standard basis theorem for quantum linear groups, Adv. in Math. 102 (1993), 202–229.CrossRefGoogle Scholar
5.Kelly, A., Lenagan, T.H. and Rigal, L.. Ring theoretic properties of quantum grassmannians. J. Algebra Appl. 3 (2004), 9–30.CrossRefGoogle Scholar
6.Krause, G. and Lenagan, T.H., Growth of algebras and Gelfand-Kirillov dimension. Revised edition. Graduate Studies in Mathematics, 22 (American Mathematical Society, Providence, RI, 2000).CrossRefGoogle Scholar
7.Krob, D. and Leclerc, B., Minor identities for quasi-determinants and quantum determinants. Comm. Math. Phys. 169 (1995), 1–23.CrossRefGoogle Scholar
8.Lakshmibai, V. and Reshetikhin, N.. Quantum deformations of SL_n/B and its Schubert varieties, Special functions (Okayama, 1990), 149–168, ICM-90 Satell. Conf. Proc. (Springer-Verlag, 1991).Google Scholar
9.Lenagan, T.H. and Rigal, L., The maximal order property for quantum determinantal rings, Proc. Edinburgh Math. Soc. (2) 46 (2003), 513–529.CrossRefGoogle Scholar
10.Lenagan, T.H. and Rigal, L., Quantum graded algebras with a straightening law and the AS-Cohen-Macaulay property for quantum determinantal rings and quantum grassmannians, J. Algebra 301 (2006), 670–702.CrossRefGoogle Scholar
11.McConnell, J.C. and Robson, J.C., Noncommutative Noetherian rings, Revised edition. Graduate Studies in Mathematics, vol. 30 (American Mathematical Society, Providence, RI, 2001).CrossRefGoogle Scholar
12.Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano, Lecture Notes in Mathematics, No. 808 (Springer-Verlag, 1980).CrossRefGoogle Scholar
13.Parshall, B. and Wang, J., Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439.Google Scholar
14.Rigal, L., Normalité de certains anneaux déterminantiels quantiques, Proc. Edinburgh Math. Soc. (2) 42 (1999), 621–640.CrossRefGoogle Scholar
15.Zhang, J.J., Connected graded Gorenstein algebras with enough normal elements, J. Algebra 189 (1997), 390–405.CrossRefGoogle Scholar