## Semigroups and weights for group representations

HTML articles powered by AMS MathViewer

- by Mohan S. Putcha
- Proc. Amer. Math. Soc.
**128**(2000), 2835-2842 - DOI: https://doi.org/10.1090/S0002-9939-00-05464-2
- Published electronically: March 2, 2000
- PDF | Request permission

## Abstract:

Let $G$ be a finite group. Consider a pair $\chi =(\chi _+,\chi _-)$ of linear characters of subgroups $P,P^-$ of $G$ with $\chi _+$ and $\chi _-$ agreeing on $P\cap P^-$. Naturally associated with $\chi$ is a finite monoid $M_\chi$. Semigroup representation theory then yields a representation $\theta$ of $G$. If $\theta$ is irreducible, we say that $\chi$ is a weight for $\theta$. When the underlying field is the field of complex numbers, we obtain a formula for the character of $\theta$ in terms of $\chi _+$ and $\chi _-$. We go on to construct weights for some familiar group representations.## References

- 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**, DOI 10.1090/pspum/047.1/933373 - 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** - J. J. Corliss,
*Upper limits to the real roots of a real algebraic equation*, Amer. Math. Monthly**46**(1939), 334–338. MR**4** - A. H. Clifford and G. B. Preston,
*Algebraic theory of semigroups*, Vol. 1, AMS Surveys No. 7, 1961. - 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** - 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, Inc.), New York-London, 1962. MR**0144979** - 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**, DOI 10.1007/978-1-4612-0979-9 - 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** - George Lusztig,
*Characters of reductive groups over a finite field*, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR**742472**, DOI 10.1515/9781400881772 - Mohan S. Putcha,
*A semigroup approach to linear algebraic groups*, J. Algebra**80**(1983), no. 1, 164–185. MR**690712**, DOI 10.1016/0021-8693(83)90026-1 - Mohan S. Putcha,
*Sandwich matrices, Solomon algebras, and Kazhdan-Lusztig polynomials*, Trans. Amer. Math. Soc.**340**(1993), no. 1, 415–428. MR**1127157**, DOI 10.1090/S0002-9947-1993-1127157-2 - Mohan S. Putcha,
*Classification of monoids of Lie type*, J. Algebra**163**(1994), no. 3, 636–662. MR**1265855**, DOI 10.1006/jabr.1994.1035 - Mohan S. Putcha,
*Complex representations of finite monoids*, Proc. London Math. Soc. (3)**73**(1996), no. 3, 623–641. MR**1407463**, DOI 10.1112/plms/s3-73.3.623 - 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**, DOI 10.1090/S0002-9947-1993-1091231-X

## Bibliographic Information

**Mohan S. Putcha**- Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
- Email: putcha@math.ncsu.edu
- Received by editor(s): November 1, 1998
- Published electronically: March 2, 2000
- Communicated by: Ronald M. Solomon
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1691001