Abstract
We prove that the transition matrix between a special Poincaré-Birkhoff-Witt (PBW) basis and the semicanonical basis of U +(\(\mathfrak{s}\mathfrak{l}_n \)(ℂ)) is upper triangular and unipotent under any order which is compatible with the partial order ⩽deg.
Similar content being viewed by others
References
Baumann P, Kamniter J. Preprojective algebras and MV polytopes. Represent Theory, 2012, 16: 152–188
Bongartz K. Minimal singularities for representations of Dykin quivers. Comment Math Helvetici, 1994, 69: 575–611
Bongartz K. On degenerations and extensions of finite-dimensional modules. Adv Math, 1996, 121: 245–287
Geiöer J. Semicanonical bases and preprojective algebras. Ann Sci Éc Norm Super, 2005, 38: 193–253
Kashiwara M. On the crystal bases of the q-analogue of universal enveloping algebras. Duke Math J, 1991, 63: 465–516
Kashiwara M, Saito Y. Geometric construction of crystal bases. Duke Math J, 1997, 89: 9–36
Lusztig G. Canonical bases arising from quantized enveloping algebras. J Amer Math Soc, 1990, 3: 447–498
Lusztig G. Canonical bases arising from quantized enveloping algebras, II. Progr Theoret Phys Suppl, 1990, 102: 175–201
Lusztig G. Quivers, perverse sheaves, and qunantized enveloping algebras. J Amer Math Soc, 1991, 4: 365–421
Lusztig G. Semicanonical bases arising from enveloping algebras. Adv Math, 2000, 151: 129–139
Ringel C M. PBW-bases of quantum groups. J Reine Angew Math, 1996, 470: 51–88
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yin, H., Zhang, S. The transition matrix between PBW basis and semicanonical basis of U +(\(\mathfrak{s}\mathfrak{l}_n \)(ℂ)). Sci. China Math. 57, 1427–1434 (2014). https://doi.org/10.1007/s11425-014-4804-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-014-4804-4