Séminaire d'Algèbre Paul Dubreil Proceedings, Paris 1976–1977 (30ème Année)

370 Pages·1978·6.44 MB·French

by  Gert Almkvist

Lecture Notes ni
Mathematics
Edited yb .A Dold dna .B nnamkcE
146
Seminaire erbeglA'd
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Proceedings, Paris 1976-1977
(30eme Annee)
Edit6 par M. .P Malliavin
galreV-regnirpS
Berlin Heidelberg New kroY 8791
Editor
Marie-Paule Malliavin
Universite Pierre et Marie Curie
10, rue Saint Louis en l'lle
75004 Paris, France
AMS Subject Classifications (1970): 12H20, 13D20, 13F20, 13G05,
13H20, 14K20, 16L20, 16A02, 16A26, 16A46, 16A60, 16A62, 16A66
16A72, 17 B20, 18 H15, 20C20, 22 E20
ISBN 3-540-08665-X Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-08665-X Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1978
Printed ni Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
2141/3140-543210
Liste des Auteurs
G. Almkvist p. - I G. Barou p. 252 - J.C. Mc Connel p. 189 - F. Couchot p. 198 -
R. Fossum p. - G. 1 Krause p. 209 - L. Lesieur p. 220 -
A. Levy-Bruhl-Laperri~re p. 163 - U. Oberst p. 112 - M. Paugam p, 298 -
H. Popp p. 281 - G, Procesi p. 128 - J. Querr6 p. 358 - H, Rahbar-Rochandel p. 339
I. Reiner p. 145 - E. Wexler-Kreindler p. 235,
TABLE DES MATIERES
G. ALMKVIST et R. FOSSUM
Decomposition of exterior and symmetric powers of indecomposable
I/p~-modules in characteristic p and relations to invariants
U. OBERST
The use of representations in the invariant theory of not
necessarily reductive groups 211
.C PROCESI
Les Bases de Hodge dans la th6orie des invariants 821
I. REINER
Integral representations of finite groups 541
A. LEVY-BRUHL-LAPERRIERE
Spectre du de Rham Hodge sur l'espace projectif complexe 163
J.C. Mc CONNEL
The global dimension of rings of differential operators 189
.F COUCHOT
Sous-modules purs et modules de type cofini 198
G. KRAUSE
Some recent developments in the theory of noetherian rings 209
L. LESIEUR
Conditions noeth@riennes dans l'anneau de polynSmes de
Ore A [X,r ,~] 220
E. WEXLER-KREINDLER
Propri@t6s de transfert des extensions d~Ore 235
G. BAROU
Cohomologie locale des alg~bres enveloppantes d'Alg~bres de
Lie nilpotentes 252
H. POPP
Recent developments in the classification theory of algebraic
varieties 281
M. PAUGAM
Sur les invariants homologiques des anneaux locaux noetheriens :
un calcul de ]a cinqui~me d@flection ~5 298
H. RAHBAR-ROCHANDEL
Relations entre la s@rie de Betti d'un anneau local de Gorenstein
et celle de l'anneau R/Socle R 339
J. QUER~
Intersections d'anneaux int~gres (II) 358
III
DECOMPOSITION OF EXTERIOR AND SYMMETRIC POWERS OF
INDECOMPOSABLE Z/pZ-MODULES IN CHARACTERISTIC
p AND RELATIONS TO INVARIANTS
Gert Almkvist (Lund) and
Robert Fossum (Copenhagen/Urbana).
This survey represents the extent of our workon the decomposition
of exterior and symmetric powers of indecomposable ~/pm~-modules in
characteristic p, the relations of these decompositions to invariant
theory, relations to combinatorial theory and suggestions for future
investigation. It is a neighborhood of a lecture given by the second
author at S~minaire d'Alg~bre Paul Dubreil at
l'Institut Henri Poincar~ in January, 1977. Therefore it contains
many more results and complete details of proofs.
The research was started when the second author, together with
Griffith tried to prove that the action of ~/p~ on the power series
ring k~X O .... ,Xn~ , with char k=p gave a factorial ring of invari-
ants. Because the decomposition of the graded components could be
easily calculated (see chapter 111,3), the =lass group could be
calculated and then it was shown to be zero. Thus it was assumed
that similar techniques could the used in general. But first the
decompositions, that is the structure of the homogeneous components
should be calculated. The table in 111.4 was calculated by hand (over
many cups of coffee in Treno's in Urbana) and did not help to
discover the general patterm.ln a letter to Almkvist in late 1975,
Fossum posed the problem of decomposition. Immediately Almkvist
solved the problem, using the fact that the representation ring is a
2
~-ring (which it isn't). But enough of the techniques of ~ -ring
theory can be used to push through the decompositions for ~./p~. In
the summer of 1976 Stanley wrote to Fossum that the decompositions
seemed to involve coefficients of Gaussian polynomials. This sugges-
ted further comparisons, which resulted in the general Valby Bodega
theorem which allows the change of basis in the representation ring
and thus permits the calculation of the number of components of a
given dimension that appear in a decomposition from the coefficients
of the Gaussian polynomials. Further months of calculations by the
first author has led to the many interesting relations centered on
the Hilbert series of the ring of invariants.
In what follows we give an outline of the contents of these notes,
chapter by chapter.
Chapter I. In this chapter the basic concepts are introduced. The
indecomposable k.~/pmE-modules are determined (here, as always, k
is a field of characteristic ~ p O) and the representation ring
R ~/pm~ is defined and studied. In our mind the most usefuld result
in this chapter is the Valby Bodega theorem (Proposition 1.1.7.) that
relates decompositions to Adam's operations (see also Problem VI.3.9)
in the representation ring R ~/p~. Also the several isomorphic
representations are given in the second section.
Chapter II. This is the chapter that contains what we need from
the classical theory of representations of the symmetric group. There
is a meta-theorem (Proposition 11.2.3) that relates elements in the
representation ring to exact sequences, and this is a key in going
from characteristic zero to characteristic p~ O. We discuss ~-rings
and the various families of symmetric functions. In the last section
we define and give what properties are needed of the homogeneous
Gaussian polynomials.
Chapter III. This is the main chapter, in which we demonstrate
the decompositions of the exterior and symmetric powers of the
indecomposable g/pg-modules. In chapter I we defined generalized
binomial coefficients of indecomposable ~/p~-modules ° In this
chapter we show :
Vn
Ar(Vn ) = ( V )
r
3
V
and sr(Vn+|) = ( n+r~ , for 4 0 r+n+| ~ p
Vr
where V is an indecomposable k ~./p~-module of dimension n. We
n
include a table that illustrates the decompositions.
Chapter IV. In this chapter we repeat, for the reader's benefit,
the calculations that show that the rings of invariants are generally
not Cohen-Macaulay. This involves calculation of a principal
homogeneous bundle.
Chapter V. This chapter, the longest and most difficult, is
devoted to the study of the dimension of the homogeneous components of
the rings of invariants. We first define the Hilbert series, provide
some examples and begin the calculations . Then the series "for
large p" are discussed. The Hilbert series for small dimensional
representations are calculated, as well as those for large dimensio-
nal representations. In one section Fourier series and integrals
are used to express these Hilbert series. Results concerning counting
of partitions are obtained. And counter-examples to a conjecture of
Stanley concerning the Hilbert series of factorial rings are menti-
oned.
Chapter VI. Examples and problems conclude this survey.
A list of notation used precedes the list of references. However
we list here those notations that are not introduced. Always p denotes
a prime integer. As almost all ~heorems are true for all primes, we do
not distinguish between the even and odd prime integers. We just note
here that if a theorem is not true for p=2, it is quite obvious
The field k will be understood to have characteristic p.
Throughout the paper we denote the cyclic group ~./pm~ by %m
with a generator ~ , and written multiplicatively. The other
notation, if not standard, is found in the list of notation.
References to a result within a chapter are of the form m.n ;
m, n~ while a reference in chapter III to result m.n in
chapter I is written l.m.n. The bibliography or list of references is
arranged alphabetically by author and then by year of publication.
References are of the form [Gauss (1777~ to indicate the author and
the year of publication. If there are two papers in the same year
they are indicated by letters ]777 a, 1777 b, etc.
We would like to thank all of those who have contributed in one
4
way or another to this work. A. Melin, T. Claesson, P. Griffith,
H-B. Foxby, R. Stanley, H. Diamond, C. Curtis and I. Reiner have
given hints and suggestions along the way. Many people have listened
to various versions of some of this work and have offered sugges-
tions that have been helpful. Also K~benhavns Universitets matema-
risk institut was kind enough to invite Fossum to Copenhagen for
one year at a time when the work in this are a was most active, and
therefore he was able to communicate very efficiently with Almkvist
We both thank our respective university for encouragement (Lund and
Illinois). Fossum has been supported during the summers by the
United States National Science Foundation. And a portion of his
visit to Denmark was supported by the Danish Statens Naturvidenska-
belige Forskningsr~d. He appreciates this support.
Finally we wish to thank Professor M.P. Malliavin who suggested
this survey and kept asking for a manuscript. Thus we had to stop
finding new Hilbert series and decompositions and had to start
writing what we know.
This material is connected to many diverse areas of mathematics ....
many with which we are not familiar. For example it has been sugges-
ted, and there are many indications that it might be true, that
there is a close connection between these decompositions and
representations of the symmetric groups in characteristic p ~ o. We
apologize to those whose results we have inadvertently rediscovered
But we are also interested in learning of other work that is closely
connected with these results.
Table of contents
O. Introduction
I. Indecomposable ~/pm~-modules and the representation ring
.] Indecomposable representations and the reprentation ring.
2. Bases for representations.
II. Representations of the symmetric group in characteristic zero.
.] Partitions, representations and symmetric functions.
2. Schur functions.
3. ~ -operations and ~ -rings.
4. Gaussian polynomials and symmetric functions.
III. Decompositions
I. The decomposition of exterior powers.
2. The decomposition of symmetric powers.
3. The decomposition of symmetric powers of V m .
P
5
4. Tables.
IV. The geometry of the group action.
.I The rings S" (Vn+]) ~pm are usually not Cohen-Macambly.
2. These ring are factorial.
3. Related results.
V. Number of invariants and Hilbert series.
.I Hilbert series and Molien'stheorem.
2. The number of invariants when p is large.
3. Computation of the Hilbert series for n=|, 2, 3, 4.
4. Fourier series and definite integrals ; a formula for
Hi(S" (Vn+l)PP) .
5. Symmetry of the Hilbert series ; a conjecture of Stanley.
VI. Examples and problems.
.I Examples in small dimensions.
2. Bertin's example.
3. Problems.
VII. Notation ,
VIII. References
Gert AlmRvist (Lund/Sverige) Robert Fossum (K~benhavn/Danmare)
Norges grunnlovsdag 1977
"It you can't stand your analyst, see your local algebraist" GA 1976.
I. INDECOMPOSABLE z/pmz-MODULES AND THE REPRESENTATION RING
.I Indecomposable representations and the representation rin~.
A representation of ~ m over k is a finite dimensional
P
vector space V over k together with a group homomorphism
Ppm 7 G Lk(V). This is the same as to say that V is a finitely
generated module over the group ring k~ m . The representation is
P
indecomposabl e if it is not the direct sum of two k~pm-modules. It
is irreducible if there is no proper k~ m-submodule.
P
m
Proposition l.l. a) the group ring k~ m~k[T]/(T-l) p kIT].
p
b) If V is an indecomposable representation, then
V~ k[T]/(r-l) n e[T~ where n = dim e V, ~ 1 n~ m p , and each
V n:= e IT] / (T-I) n k IT] si~ indecomFosable.
c) The indecomposable V m is both free and
p
6
in~ective s~__a a k~ m-module.
P
d) The only irreducible ))k m-module (up to
P
isomorphism) is V]~ k.
Proof. a) The group ring k~ m is generated as a k-algebra by ~ .
P
Define kiT| >k~ m by extending T - - ~ . Since char k = p
P
and ~pm = ,| the element (T-I)P m is in the kernel. Hence there is
a surjection k[T]/(T-I)P m k[T] >k~ m . Comparing dimensions over
P
k yields that it is an isomorphism.
pm
c) The ring k[r]/(r-l) k[T~ is a local quasi-frobenius at-
tin k-algebra with maximal ideal generated by the image of T-I
Hence V m : = k~ m is free (obvious) and injective.
P P
d) It is clear that k~k[r]/(T-l) kIT| is irreducible as
a k p) m-module. Suppose V is a finite dimensional k~p m-module.
Then the soele of V, by definition Sot(V) : = HOmk~pm (k,V), is
a k/ m-submodule. If V is irreducible, then V = Sot(V). But
P
Sot(V) = (dim k Soc(V)).V 1 as k~pm-modules. Hence
dim k Sot(V) = dim k V = .I
b) We prove slightly more than b). In fact we prove that a
k~pm-module V decomposes into as many indeeomposables as
dim k Sot(V). First, since k))pm is a local ring and artinian,
any cyclic module (i.e. one of the form kP m/6~. ) is indecomposable.
P
Each ideal is of the form (T-I) n k~m and hence each V is
n
indecomposable. Now Soe(V n) = (T-I) n-| Vn and is one dimensional.
As Soc (V) >V is essential whenever V is of finite type, the
injective enveloppe of V is determined by the injective enveloppe
of its socle. As VI~ Soc (kppm) .~ 2- k~ep m is essential and
k~pm is injective and indecomposable, it is seen that E(V l) Vpm.
Suppose there is an injection VI~ > V. Let V(V )l denote the
maximal essential extension of V in V. Then the claim is that
1
V(V )I is a direct summand of V, Suppose W is maximal in V
with respect to the property that W~V(VI) = 0. (This is the same
as to say that WNV 1 = 0). Then the composition V(VI)C-->V----~V/W
is an injection. Furthermore it is essential, since Soc(V/W)~V|.
By the maximality of V(V )I it is a surjection. Hence V(V )I is a
direct summand.
As a corollary, the module V is indeeomposable if and only
if Soc(V)~V .I But then there is an embedding V >k~pm = E(VI).
Hence V = (T-l) r k~ m. But (T-l) r k~ m = V m
p p p -r
QED.
In the last paragraph above we have used the fact that the
k-linear dual of a ~pm-representation is isomorphic to the original
representation. For as ~ m-modules there is an isomorphism
P
J~
kppm = Homk(kppm , k)
and hence
H°mk>~ m (V, k~pm)-- -~ HOmk(V,k)
P
for each k~ m-module V. Then it follows that
P
HOmk(Vn,k) =~V n
as k~ m-modules.
P
The representation ring of k~ m is defined to be the free
P
abelian group on the isomorphism classes ~] of k~ m-modules of
P
finite type, modulo the relations ~] = ~'] + [V"] provided
V ~V' • V". Denote this abelian group by RkP m.
P
Corollary 1.2. : The abelian group Rk~ m is free on the elements
P
Vl,...,Vpm •
DEQ
The ring structure in Rk)~ m is induced by . k O So
V.W ; = V k O W, (We omit any kind of symbols to denote the classes
of a representation V in Rk~ m. And we interchange freely the
P
notation V.W and V @k W for the product. This should cause no
confusion. Likwise V+W means V O W as modules or
"~] + [W]" in Rk~ m).
P
Proposition 1.3. : s~A ~ ~-al~ebra, the rin~ Rk~pm s~_i generated by
Vpo+l , Vpl+|, Vp2+l,''',Vpm-l+l
A proof of this proposition depend~ upon obtaining the
decomposition of the tensor products e V ® V m. This is not done in
this paper. However the multiplication table below, which is found
in [Rally (1969)] permits us to demonstrate the proposition.
(The history of the decomposition of V~ m ~ V is not clear
to us. It seems that Littlevood knew the decomposition constants.
Also Green, Srinivasan, and Rally have discussed them. That the
Vpi+| generate the representation algebra is explicity mentioned
in [Srinivasan (1964)~ See also the papers by Renaud.)