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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**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**, 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****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****5.**Ton That Tri,*The irreducible modular representations of semigroups of all matrices*, Acta Math. Vietnam.**20**(1995), no. 1, 43–53. MR**1346347****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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
20C20

Retrieve articles in all journals with MSC (1991): 20C20

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:
https://doi.org/10.1090/S0002-9939-99-05424-6

Received by editor(s):
April 10, 1998

Published electronically:
September 9, 1999

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1999
American Mathematical Society