Geodesic
QUASI-DETERMINANTS AND q-COMMUTING MINORS
Glasgow mathematical journal, Tome 52 (2010) no. 3, pp. 663-675

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.1017/S0017089510000509
Mots-clés : 20G42, 16T30, 15A15 
LAUVE, AARON. QUASI-DETERMINANTS AND q-COMMUTING MINORS. Glasgow mathematical journal, Tome 52 (2010) no. 3, pp. 663-675. doi: 10.1017/S0017089510000509
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[1] 1.Berenstein, A. and Zelevinsky, A., String bases for quantum groups of type A, In I. M. Gel′fand Seminar, vol. 16, Adv. Soviet Math. (American Mathematical Society, Providence, RI, 1993), 51–89. Google Scholar | DOI

[2] 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. Google Scholar | DOI

[3] 3.Fioresi, R., Quantum deformation of the flag variety, Commun. Algebra 27 (11) (1999), 5669–5685. Google Scholar | DOI

[4] 4.Gelfand, I., Gelfand, S., Retakh, V. and Wilson, R. L., Quasideterminants, Adv. Math. 193 (1) (2005), 56–141. Google Scholar

[5] 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] 6.Goodearl, K. R., Commutation relations for arbitrary quantum minors, Pacific J. Math. 228 (1) (2006), 63–102. Google Scholar | DOI

[7] 7.Goodearl, K. R. and Lenagan, T. H., Quantum determinantal ideals, Duke Math. J. 103 (1) (2000), 165–190. Google Scholar | DOI

[8] 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] 9.Kelly, A. C., Lenagan, T. H. and Rigal, L., Ring theoretic properties of quantum Grassmannians, J. Algebra Appl. 3 (1) (2004), 9–30. Google Scholar | DOI

[10] 10.Krob, D. and Leclerc, B., Minor identities for quasi-determinants and quantum determinants, Commun. Math. Phys. 169 (1) (1995), 1–23. Google Scholar | DOI

[11] 11.Lakshmibai, V. and Reshetikhin, N., Quantum deformations of SL 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] 12.Lauve, A., Quantum- and quasi-Plücker coordinates, J. Algebra 296 (2) (2006), 440–461. Google Scholar | DOI

[13] 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] 14.Scott, J., Quasi-commuting families of quantum minors, J. Algebra 290 (1) (2005), 204–220. Google Scholar

[15] 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] 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

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