### Abstract

Original language | English |
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Pages (from-to) | 497-510 |

Number of pages | 14 |

Journal | The Quarterly Journal of Mathematics |

Volume | 61 |

Issue number | 4 |

Early online date | 2 May 2009 |

DOIs | |

Publication status | Published - Dec 2010 |

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

Research output: Contribution to journal › Article

*The Quarterly Journal of Mathematics*, vol. 61, no. 4, pp. 497-510. https://doi.org/10.1093/qmath/hap016

}

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 -