# ON A THEOREM OF BARBARA SCHMID
ON A THEOREM OF BARBARA SCHMID
ON A THEOREM OF BARBARA SCHMID
ON A THEOREM OF BARBARA SCHMID
ON A THEOREM OF BARBARA SCHMID
LARRY SMITH
LARRY SMITH
LARRY SMITH
LARRY SMITH
LARRY SMITH
# AND
SCHOOL OF MATHEMATICS
UNIVERSITY OF MINNESOTA
ORDWAY LECTURE
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : Let G be a finite group and # : G # GL(n, C) a complex representation. Barbara
Schmid has shown that the algebra of invariant polynomial functions C[V ] G on the vector
space V = C n is generated by homogeneous polynomials of degree at most b, where b is the
largest degree of a generator in a minimal generating set for C[reg C (G)] G , and reg C (G) is the
complex regular representation of G. In this note we give a new proof of this result, and at
the same time extend it to fields IF whose characteristic p is larger than ïGï, the order of the
group G.
MATHEMATICS SUBJECT CLASSIFICATION : 13A50 Invariant Theory
Typeset by LS T E X
Let # : G # GL(n, IF) be a representation of a finite group G over the field IF. Then G acts
on the vector space V = IF n , and hence also on the algebra IF[V ] of polynomial functions
on V . The algebra fixed by this action is called the ring of invariants, and is denoted by
IF[V ] G . Note that IF[V ] G depends on #, but # does not appear in the notation. Despite this
no confusion should arise.
By theorems of D. Hilbert [1] and E. Noether [2], [3] it is known that IF[V ] G is finitely
generated as an algebra over IF (see also [5], Chapter 2, to which we refer for basic facts
from invariant theory). The maximal degree of a generator in any minimal generating set
for IF[V ] G is denoted by b(#). This is nothing but the degree of the Poincar’ e polynomial
P(QIF[V ] G , t) of the module of indecomposable elements
QIF[V ] G = IF Ä IF[V ] G IF[V ] G
= IF[V ] G
/ (IF[V ] G
) 2 ,
where IF[V ] G
denotes the augmentation ideal of IF[V ] G . (See for example [5] Chapter 4
and Section 5.1) In [4] Barbara Schmid proved, among other things, that for a complex
representation # : G # GL(n, C)
b(#) £ b(reg C (G)) ,
where reg C (G) : G # GL(ïGï, C) is the complex regular representation of G. The purpose
of this note is to provide what I feel is a more conceptual proof of this theorem, which, at the
same time extends the result to the strong nonmodular case [7]. Specifically we will prove:
THEOREM THEOREM THEOREM THEOREM THEOREM : Let G be a finite group of order d and IF a field of characteristic p. If p > d then
b(#) £ b(reg IF (G)) for any representation # : G # GL(n, IF), where reg IF (G) : G # GL(d, IF)
is the regular representation of G over the field IF.
This research was done in the course of preparing a series of lectures on the invariant theory
of finite groups which were presented at the University of Minnesota School of Mathematics
in the spring quater of 1998 when the author was an Ordway Visitor. I would like to thank
the participants in the lecture series for their atttentive, critical attitude, and their often
penetrating questions, and the School of Mathematics for providing an atmosphere conducive
to productive research and study.
The proof of the theorem relies on the main result of [8] (see also [5] Theorem 3.1.10) and
the following Proposition. Taken together I think these results explain conceptually why the
regular representation provides a universal upper bound for b(#).
PROPOSITION PROPOSITION PROPOSITION PROPOSITION PROPOSITION : Let IF be a field, G be a finite group, and W a finite transitive Gset. Then
W occurs as an orbit in the regular representation of G defined over IF.
PROOF PROOF PROOF PROOF PROOF : Identify the regular representation of G over IF with the action of G from the left
on the group ring IF(G) of G over IF. If H £ G is the isotropy group of a point of W then we
may identify W as a Gset with the left action of G on the left cosets of H in G, so W ~ = G / H
as Gsets. Let
rH = X hÎH
h ÎIF(G).
Then the isotropy group of r H in IF(G) is H , so the orbit of r H is also isomorphic to G / H as a
left Gset.
LARRY SMITH
PROOF OF THE THEOREM
PROOF OF THE THEOREM
PROOF OF THE THEOREM
PROOF OF THE THEOREM
PROOF OF THE THEOREM : Since d =ïGï> p it follows from [8] (see also [5] Theorem 3.1.10)
that IF[V ] G is generated by orbit Chern classes. Write V * for the space of linear forms on V .
If B # V * is an orbit of G, then, by the Proposition, we may find a G equivariant embedding
B # IF(G). This in turn induces an epimorphism
a B : IF[IF(G)] # IF[B] ,
where IF[B] is the polynomial algebra in the formal variables b ÎB . Since p # d passing to
fixed subalgebras yields an epimorphism
a G
B : IF[IF(G)] G
## IF[B] G .
If k denotes the number of elements in B, then, the action of G on B, being by permutations,
leaves the elementary symmetric polynomials e 1 , . . . , e k in the elements b ÎB invariant, i.e.,
IF[e 1 (B) , . . . , e k (B)] # IF[B] G .
The inclusion B # V * induces a Gequivariant map
b B : IF[B] ## IF[V ] ,
and, per definition [8],
b B (e i (B)) = c i (B) i = 1 , . . . , k ,
where c 1 (B) , . . . , c k (B) ÎIF[V ] G are the Chern classes of the orbit B. Therefore the composite
c B : IF[IF(G)] G a G
B
# IF[B] G b G
B
# IF[V ] G
contains the characteristic subalgebra of the orbit B in its image, i.e., Im(c B ) contains the
subalgebra of IF[V ] G generated by the orbit Chern classes c 1 (B) , . . . , c k (B) ÎIF[V ] G of the
orbit B. Since IF[V ] G is generated by orbit Chern classes, this means we can find a finite
number B 1 , . . . , Bm of orbits of G in V * such that the map
m
Ä
i=1
: ÄIF[IF(G)] G Ä m
i=1 c B i
# m
Ä
i=1
IF[V ] G l
# IF[V ] G ,
is an epimorphism, where l is the multiplication map. Since
m
Ä
i=1
IF[IF(G)] G is generated by
forms of degree at most b IF (reg IF (G)) the result follows.
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ON A THEOREM OF BARBARA SCHMID
References
[1] D. Hilbert, ˜
Uber die Theorie der Algebraischen Formen, Math. Ann. 36 (1890),
473--534.
[2] E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77
(1916), 89--92.
[3] E. Noether, Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Char
acteristik p, Nachr. v. d. Ges. d. Wiss. zu G˜ ottingen (1926), 28--35.
[4] B. J. Schmid, Finite Groups and Invariant Theory, S’ eminaire d'Alg‘ ebre P. Dubriel et
M.P. Malliavin 19891990, Lecture Notes in Math. 1478, SpringerVerlag, Heidelberg,
Berlin, 1991.
[5] L. Smith, Polynomial Invariants of Finite Groups, (second printing), A.K. Peters Ltd.,
Wellesley, MA 1995, 1997.
[6] L. Smith, Noether's Bound in the Invariant Theory of Finite Groups, Arch. der Math.
66 (1996), 89 -- 92.
[7] L. Smith, Polynomial Invariants of Finite Groups, A Survey of Recent Results, Bull. of
the Amer. Math. Soc. 34 (1997), 211 -- 250.
[8] L. Smith and R. E. Stong, On the Invariant Theory of Finite Groups: Orbit Polynomials
and Splitting Principles, J. of Algebra 110 (1987), 134--157.
Larry Smith
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
email: SMITH@MATH.UMN.EDU
and
# Mathematisches Institut der Universit ˜
at
D 37073 G˜ ottingen
Federal Republic of Germany
email: LARRY@SUNRISE.UNIMATH.GWDG.DE
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