Representation Theory of Reductive Normal Algebraic Monoids

Doty, Stephen

doi:10.1090/S0002-9947-99-02462-9

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Representation Theory of
Reductive Normal Algebraic Monoids

Author: Stephen Doty
Journal: Trans. Amer. Math. Soc. 351 (1999), 2539-2551
MSC (1991): Primary 20G05, 20M30; Secondary 16G99, 22E55
Posted: February 15, 1999
MathSciNet review: 1653351
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Abstract: New results in the representation theory of ``semisimple'' algebraic monoids are obtained, based on Renner's monoid version of Chevalley's big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as ``polynomial'' representations of the associated reductive group of units in the monoid, and this representation category splits into a direct sum of subcategories by ``homogeneous'' degree. We show that each of these homogeneous subcategories is a highest weight category, in the sense of Cline, Parshall, and Scott, and so equivalent with the module category of a certain finite-dimensional quasihereditary algebra, which we show is a generalized Schur algebra in S. Donkin's sense.

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