Semigroups and weights for group representations

Author:
Mohan S. Putcha

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2835-2842

MSC (2000):
Primary 20C99, 20M30

DOI:
https://doi.org/10.1090/S0002-9939-00-05464-2

Published electronically:
March 2, 2000

MathSciNet review:
1691001

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Let be a finite group. Consider a pair of linear characters of subgroups of with and agreeing on . Naturally associated with is a finite monoid . Semigroup representation theory then yields a representation of . If is irreducible, we say that is a weight for . When the underlying field is the field of complex numbers, we obtain a formula for the character of in terms of and . We go on to construct weights for some familiar group representations.

**1.**J. L. Alperin,*Weights for finite 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. 369–379. MR**933373****2.**Roger W. Carter,*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307****3.**A. H. Clifford,*Matrix representations of completely simple semigroups*, Amer. J. Math.**64**(1942), 327-342. MR**4:4a****4.**A. H. Clifford and G. B. Preston,*Algebraic theory of semigroups*, Vol. 1, AMS Surveys No. 7, 1961.**5.**C. W. Curtis,*Modular representations of finite groups with split (𝐵,𝑁)-pairs*, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Springer, Berlin, 1970, pp. 57–95. MR**0262383****6.**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR**0144979****7.**William Fulton and Joe Harris,*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249****8.**Gordon James and Adalbert Kerber,*The representation theory of the symmetric group*, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR**644144****9.**George Lusztig,*Characters of reductive groups over a finite field*, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR**742472****10.**Mohan S. Putcha,*A semigroup approach to linear algebraic groups*, J. Algebra**80**(1983), no. 1, 164–185. MR**690712**, https://doi.org/10.1016/0021-8693(83)90026-1**11.**Mohan S. Putcha,*Sandwich matrices, Solomon algebras, and Kazhdan-Lusztig polynomials*, Trans. Amer. Math. Soc.**340**(1993), no. 1, 415–428. MR**1127157**, https://doi.org/10.1090/S0002-9947-1993-1127157-2**12.**Mohan S. Putcha,*Classification of monoids of Lie type*, J. Algebra**163**(1994), no. 3, 636–662. MR**1265855**, https://doi.org/10.1006/jabr.1994.1035**13.**Mohan S. Putcha,*Complex representations of finite monoids*, Proc. London Math. Soc. (3)**73**(1996), no. 3, 623–641. MR**1407463**, https://doi.org/10.1112/plms/s3-73.3.623**14.**Mohan S. Putcha and Lex E. Renner,*The canonical compactification of a finite group of Lie type*, Trans. Amer. Math. Soc.**337**(1993), no. 1, 305–319. MR**1091231**, https://doi.org/10.1090/S0002-9947-1993-1091231-X

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Additional Information

**Mohan S. Putcha**

Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Email:
putcha@math.ncsu.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05464-2

Received by editor(s):
November 1, 1998

Published electronically:
March 2, 2000

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 2000
American Mathematical Society