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), 401405
MSC (1991):
Primary 20C20
Published electronically:
September 9, 1999
MathSciNet review:
1676308
Fulltext 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 SchwartzTri'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
(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,
AddisonWesley 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), 319369.
 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), 449517. CMP 99:01
 1.
 S. Doty and G. Walker, Truncated symmetric powers and modular representations of , Math. Proc. Camb Phil. Soc. 119 (1996), 231242. MR 96g:20062
 2.
 G.D. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications vol. 16 (AddisonWesley), 1981. MR 83k:20003
 3.
 H. Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sec. IA 22 (1975), 319369.
 4.
 L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lecture Notes in Mathematics, 1994. MR 95d:55017
 5.
 T. T. Tri, The irreducible modular representations of semigroups of all matrices, Acta Math. Vietnamica 20 (1995), 4353. MR 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), 449517. CMP 99:01
Similar Articles
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:
http://dx.doi.org/10.1090/S0002993999054246
PII:
S 00029939(99)054246
Received by editor(s):
April 10, 1998
Published electronically:
September 9, 1999
Communicated by:
Ronald M. Solomon
Article copyright:
© Copyright 1999
American Mathematical Society
