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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The first occurrence for the
irreducible modules of general linear
groups in the polynomial algebra


Authors: Pham Anh Minh and Ton That Tri
Journal: Proc. Amer. Math. Soc. 128 (2000), 401-405
MSC (1991): Primary 20C20
Published electronically: September 9, 1999
MathSciNet review: 1676308
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $p$ be a prime number and let $GL_{n}$ be the group of all invertible matrices over the prime field $\mathbb{F}_p$. It is known that every irreducible $GL_{n}$-module can occur as a submodule of $P$, the polynomial algebra with $n$ variables over $\mathbb{F}_p$. Given an irreducible $GL_{n}$-module $\rho $, the purpose of this paper is to find out the first value of the degree $d$ of which $\rho$ occurs as a submodule of $P_{d}$, the subset of $P$ consisting of homogeneous polynomials of degree $d$. This generalizes Schwartz-Tri's result to the case of any prime $p$.


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

  • 1. Stephen Doty and Grant Walker, Truncated symmetric powers and modular representations of 𝐺𝐿_{𝑛}, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 2, 231–242. MR 1357041 (96g:20062), http://dx.doi.org/10.1017/S0305004100074120
  • 2. 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 (83k:20003)
  • 3. H. Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sec. IA 22 (1975), 319-369.
  • 4. Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1282727 (95d:55017)
  • 5. Ton That Tri, The irreducible modular representations of semigroups of all matrices, Acta Math. Vietnam. 20 (1995), no. 1, 43–53. MR 1346347 (96f:20103)
  • 6. T. T. Tri, On a conjecture of Grant Walker for the first occurrence of irreducible modular representations of general linear groups, submitted.
  • 7. R. M. W. Wood, Problems in the Steenrod algebra, Bull. London Math. Soc. 30 (1998), 449-517. CMP 99:01

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Additional Information

Pham Anh Minh
Affiliation: Department of Mathematics, College of Sciences, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam
Email: paminh@bdvn.vnd.net

Ton That Tri
Affiliation: Department of Mathematics, College of Sciences, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05424-6
PII: S 0002-9939(99)05424-6
Received by editor(s): April 10, 1998
Published electronically: September 9, 1999
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1999 American Mathematical Society