Representation Theory of Reductive Normal Algebraic Monoids

Doty, Stephen

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

Representation Theory of Reductive Normal Algebraic Monoids
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by Stephen Doty

Trans. Amer. Math. Soc. 351 (1999), 2539-2551

DOI: https://doi.org/10.1090/S0002-9947-99-02462-9

Published electronically: February 15, 1999

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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.

References

H. D. Macpherson and Cheryl E. Praeger, Maximal subgroups of infinite symmetric groups, J. London Math. Soc. (2) 42 (1990), no. 1, 85–92. MR 1078176, DOI 10.1112/jlms/s2-42.1.85
E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
Michel Demazure and Peter Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland Mathematics Studies, vol. 39, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from the French by J. Bell. MR 563524
S. Donkin, On Schur algebras and related algebras. I, J. Algebra 104 (1986), no. 2, 310–328. MR 866778, DOI 10.1016/0021-8693(86)90218-8
S. Donkin, On Schur algebras and related algebras. I, J. Algebra 104 (1986), no. 2, 310–328. MR 866778, DOI 10.1016/0021-8693(86)90218-8
S. Donkin, Good filtrations of rational modules for reductive groups, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 69–80. MR 933351, DOI 10.1090/pspum/047.1/933351
S. R. Doty, Resolutions of $B$ modules, Indag. Math. (N.S.) 5 (1994), no. 3, 267–283. MR 1298774, DOI 10.1016/0019-3577(94)90003-5
Stephen Doty, Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), no. 1-3, 165–199. MR 1492900, DOI 10.1016/S0022-4049(96)00082-5
Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. MR 1427618, DOI 10.1007/s002220050119
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
Mohan S. Putcha, On linear algebraic semigroups. III, Internat. J. Math. Math. Sci. 4 (1981), no. 4, 667–690. MR 663652, DOI 10.1155/S0161171281000513
Mohan S. Putcha, Linear algebraic monoids, London Mathematical Society Lecture Note Series, vol. 133, Cambridge University Press, Cambridge, 1988. MR 964690, DOI 10.1017/CBO9780511600661
Lex E. Renner, Classification of semisimple rank one monoids, Trans. Amer. Math. Soc. 287 (1985), no. 2, 457–473. MR 768719, DOI 10.1090/S0002-9947-1985-0768719-7
Lex E. Renner, Conjugacy classes of semisimple elements, and irreducible representations of algebraic monoids, Comm. Algebra 16 (1988), no. 9, 1933–1943. MR 952217, DOI 10.1080/00927878808823667
L. Solomon, An introduction to reductive monoids, in Semigroups, formal languages and groups (York, 1993), 295–352, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 466, Kluwer Acad. Publ., Dordrecht, 1995.