Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. J. L. Alperin, Weights for finite groups, Proc. Symp. Pure Math. 47 (1987), 369-379. MR 89h:20015
  • 2. R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex character, Wiley (1985). MR 87d:20060
  • 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 $(B,N)$-pari, Lecture Notes in Math. 131 (1970), 57-95. MR 41:6991
  • 6. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Wiley (1962). MR 26:2519
  • 7. W. Fulton and J. Harris, Representation theory, Graduate Texts in Mathematics 129, Springer-Verlag, 1991. MR 93a:20069
  • 8. G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its applications, Vol. 16, Addison-Wesley, 1981. MR 83k:20003
  • 9. G. Lusztig, Characters of reductive groups over a finite field, Annals. of Math. Studies 107 (1984), Princeton University Press. MR 86j:20038
  • 10. M. S. Putcha, A semigroup approach to linear algebraic groups, J. Algebra 80 (1983), 164-185. MR 84j:20045
  • 11. -, Sandwich matrices, Solomon algebras and Kazhdan-Lusztig polynomials, Trans. Amer. Math. Soc. 340 (1993), 415-428. MR 94a:20112
  • 12. -, Classification of monoids of Lie type, J. Algebra 163 (1994), 636-662. MR 95b:20089
  • 13. -, Complex representations of finite monoids, Proc. London Math. Soc. 73 (1996), 623-641. MR 97e:20093
  • 14. M. S. Putcha and L. E. Renner, The canonical compactification of a finite group of Lie type, Trans. Amer. Math. Soc. 337 (1993), 305-319. MR 93g:20123

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20C99, 20M30

Retrieve articles in all journals with MSC (2000): 20C99, 20M30


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

American Mathematical Society