Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-07-05T13:47:41.618Z Has data issue: false hasContentIssue false
  • English
  • Français

QUASI-DETERMINANTS AND q-COMMUTING MINORS

Published online by Cambridge University Press:  25 August 2010

AARON LAUVE
AARON LAUVE*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA e-mail: lauve@math.luc.edu
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 present two new proofs of the q-commuting property holding among certain pairs of quantum minors of a q-generic matrix. The first uses elementary quasi-determinantal arithmetic; the second involves paths in a directed graph. Together, they indicate a means to build the multi-homogeneous coordinate rings of flag varieties in other non-commutative settings.

Keywords

20G4216T3015A15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 3 , September 2010 , pp. 663 - 675
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Berenstein, A. and Zelevinsky, A., String bases for quantum groups of type Ar, In I. M. Gel′fand Seminar, vol. 16, Adv. Soviet Math. (American Mathematical Society, Providence, RI, 1993), 51–89.CrossRefGoogle Scholar
2.Danilov, V. I., Karzanov, A. V. and Koshevoy, G. A., Plücker environments, wiring and tiling diagrams, and weakly separated set-systems, Adv. Math. 224 (1) (2010), 1–44.CrossRefGoogle Scholar
3.Fioresi, R., Quantum deformation of the flag variety, Commun. Algebra 27 (11) (1999), 5669–5685.CrossRefGoogle Scholar
4.Gelfand, I., Gelfand, S., Retakh, V. and Wilson, R. L., Quasideterminants, Adv. Math. 193 (1) (2005), 56–141.Google Scholar
5.Gel′fand, I. M. and Retakh, V. S., Determinants of matrices over noncommutative rings, Funktsional. Anal. i Prilozhen. 25 (2) (1991), 13–25, 96.Google Scholar
6.Goodearl, K. R., Commutation relations for arbitrary quantum minors, Pacific J. Math. 228 (1) (2006), 63–102.CrossRefGoogle Scholar
7.Goodearl, K. R. and Lenagan, T. H., Quantum determinantal ideals, Duke Math. J. 103 (1) (2000), 165–190.CrossRefGoogle Scholar
8.Hong, J. and Kang, S.-J., Introduction to quantum groups and crystal bases, vol. 42, Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2002).Google Scholar
9.Kelly, A. C., Lenagan, T. H. and Rigal, L., Ring theoretic properties of quantum Grassmannians, J. Algebra Appl. 3 (1) (2004), 9–30.CrossRefGoogle Scholar
10.Krob, D. and Leclerc, B., Minor identities for quasi-determinants and quantum determinants, Commun. Math. Phys. 169 (1) (1995), 1–23.CrossRefGoogle Scholar
11.Lakshmibai, V. and Reshetikhin, N., Quantum deformations of SLn/B and its Schubert varieties, in Special functions (Okayama, 1990), ICM-90 Satellite Conference Proceedings (Kashiwara, M. and Miwa, T., Editors) (Springer, Tokyo, 1991), 149–168.Google Scholar
12.Lauve, A., Quantum- and quasi-Plücker coordinates, J. Algebra 296 (2) (2006), 440–461.CrossRefGoogle Scholar
13.Leclerc, B. and Zelevinsky, A., Quasicommuting families of quantum Plücker coordinates, in Kirillov's seminar on representation theory, vol. 181, American Mathematical Society, Series (translation) 2 (Providence, RI, 1998), 85–108.Google Scholar
14.Scott, J., Quasi-commuting families of quantum minors, J. Algebra 290 (1) (2005), 204–220.Google Scholar
15.Taft, E. and Towber, J., Quantum deformation of flag schemes and Grassmann schemes, I. A q-deformation of the shape-algebra for GL(n), J. Algebra 142 (1) (1991), 1–36.Google Scholar
16.Takeuchi, M., A short course on quantum matrices, in New directions in Hopf algebras, vol. 43, Publications of the Research Institute for Mathematical Sciences, (Montgomery, S. and Schneider, H.-J., Editors) (Cambridge University Press, Cambridge, UK, 2002), 383–435. Notes taken by Bernd Strüber.Google Scholar