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The Quarterly Journal of Mathematics Advance Access published online on May 3, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap016
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

A GENERIC MULTIPLICATION IN QUANTIZED SCHUR ALGEBRAS

Xiuping Su{dagger}

Department of Mathematical Sciences, University of Bath, Bath BA2 7JY, UK

{dagger} E-mail: xs214{at}bath.ac.uk

Received 6 January 2009; revised 4 April 2009
  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 GLn, 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.


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