Semigroups and weights for group representations
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- 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
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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