A generic multiplication in quantized Schur algebras

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We define a generic multiplication in quantized Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantized Schur algebras, defined in (A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GL(n), Duke Math. J. 61 (1990), 655-677), a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in (M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147-163). We also prove that the subalgebra of the new algebra gives a geometric realization of a positive part of 0-Schur algebras, defined in (S. Donkin, The q-Schur Algebra, London Mathematical Society Lecture Note Series 253. Cambridge University Press, Cambridge, 1998, x + 179. ISBN: 0-521-64558-1.). Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.
Original languageEnglish
Pages (from-to)497-510
Number of pages14
JournalThe Quarterly Journal of Mathematics
Volume61
Issue number4
Early online date2 May 2009
DOIs
Publication statusPublished - Dec 2010

Fingerprint

Schur Algebras
Multiplication
Algebra
Quantum Groups
Subalgebra
Multiplicative
Hall Algebra
Monoid
Quotient
Series

Cite this

A generic multiplication in quantized Schur algebras. / Su, Xiuping.

In: The Quarterly Journal of Mathematics, Vol. 61, No. 4, 12.2010, p. 497-510.

Research output: Contribution to journalArticle

@article{544cdb52b1814e7ab5b5c1d406edf555,
title = "A generic multiplication in quantized Schur algebras",
abstract = "We define a generic multiplication in quantized Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantized Schur algebras, defined in (A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GL(n), Duke Math. J. 61 (1990), 655-677), a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in (M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147-163). We also prove that the subalgebra of the new algebra gives a geometric realization of a positive part of 0-Schur algebras, defined in (S. Donkin, The q-Schur Algebra, London Mathematical Society Lecture Note Series 253. Cambridge University Press, Cambridge, 1998, x + 179. ISBN: 0-521-64558-1.). Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.",
author = "Xiuping Su",
year = "2010",
month = "12",
doi = "10.1093/qmath/hap016",
language = "English",
volume = "61",
pages = "497--510",
journal = "The Quarterly Journal of Mathematics",
issn = "0033-5606",
publisher = "Oxford University Press",
number = "4",

}

TY - JOUR

T1 - A generic multiplication in quantized Schur algebras

AU - Su, Xiuping

PY - 2010/12

Y1 - 2010/12

N2 - We define a generic multiplication in quantized Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantized Schur algebras, defined in (A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GL(n), Duke Math. J. 61 (1990), 655-677), a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in (M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147-163). We also prove that the subalgebra of the new algebra gives a geometric realization of a positive part of 0-Schur algebras, defined in (S. Donkin, The q-Schur Algebra, London Mathematical Society Lecture Note Series 253. Cambridge University Press, Cambridge, 1998, x + 179. ISBN: 0-521-64558-1.). Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.

AB - We define a generic multiplication in quantized Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantized Schur algebras, defined in (A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GL(n), Duke Math. J. 61 (1990), 655-677), a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in (M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147-163). We also prove that the subalgebra of the new algebra gives a geometric realization of a positive part of 0-Schur algebras, defined in (S. Donkin, The q-Schur Algebra, London Mathematical Society Lecture Note Series 253. Cambridge University Press, Cambridge, 1998, x + 179. ISBN: 0-521-64558-1.). Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.

UR - http://www.scopus.com/inward/record.url?scp=78649271197&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1093/qmath/hap016

U2 - 10.1093/qmath/hap016

DO - 10.1093/qmath/hap016

M3 - Article

VL - 61

SP - 497

EP - 510

JO - The Quarterly Journal of Mathematics

JF - The Quarterly Journal of Mathematics

SN - 0033-5606

IS - 4

ER -