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The Quarterly Journal of Mathematics Advance Access published online on June 18, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han014
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© The author 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

SOME Z/2-GRADED REPRESENTATION THEORY

Brian J. Parshall{dagger} and Leonard L. Scott {ddagger}

Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA

{dagger} Corresponding author. E-mail: bjp8w{at}virginia.edu

Received 16 July 2007;
  Abstract

In representation theory, the existence of a Z+-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category O associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p≥n, though this grading would suffice to establish Lusztig's character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Astérique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces Z/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full Z+-grading, while being possibly much easier to find. We define the notion of a Z/2-based Kazhdan–Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan–Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan–Lusztig theories, Tôhoku Math. J. 45 (1993), 511–534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Extn-groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

{ddagger} E-mail: lls21{at}virginia.edu


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