Algorithms in Invariant Theory

201 Pages·2008·1.235 MB·English

by  Dr. Bernd Sturmfels (auth.)

W
Texts and Monographs in
Symbolic Computation
A Series of the
Research Institute for Symbolic Computation,
Johannes Kepler University, Linz, Austria
Edited by P. Paule
Bernd Sturmfels
Algorithms in Invariant Theory
Second edition
SpringerWienNewYork
Dr. Bernd Sturmfels
Department of Mathematics
University of California, Berkeley, California, U.S.A.
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Preface
The aim of this monograph is to provide an introduction to some fundamental
problems, results and algorithms of invariant theory. The focus will be on the
three following aspects:
(i) Algebraic algorithms in invariant theory, in particular algorithms arising
from the theory of Gröbner bases;
(ii) Combinatorial algorithms in invariant theory, such as the straightening al-
gorithm, which relate to representation theory of the general linear group;
(iii) Applications to projective geometry.
Part of this material was covered in a graduate course which I taught at RISC-
Linz in the spring of 1989 and at Cornell University in the fall of 1989. The
specific selection of topics has been determined by my personal taste and my
belief that many interesting connections between invariant theory and symbolic
computation are yet to be explored.
In order to get started with her/his own explorations, the reader will find
exercises at the end of each section. The exercises vary in difficulty. Some of
them are easy and straightforward, while others are more difficult, and might in
fact lead to research projects. Exercises which I consider “more difficult” are
marked with a star.
This book is intended for a diverse audience: graduate students who wish
to learn the subject from scratch, researchers in the various fields of application
who want to concentrate on certain aspects of the theory, specialists who need
a reference on the algorithmic side of their field, and all others between these
extremes. The overwhelming majority of the results in this book arewell known,
with many theorems dating back to the 19th century. Some of the algorithms,
however, are new and not published elsewhere.
I am grateful to B. Buchberger, D. Eisenbud, L. Grove, D. Kapur, Y. Laksh-
man, A. Logar, B. Mourrain, V. Reiner, S. Sundaram, R. Stanley, A. Zelevinsky,
G. Ziegler and numerous others who supplied comments on various versions of
the manuscript. Special thanks go to N. White for introducing me to the beau-
tiful subject of invariant theory, and for collaborating with me on the topics in
Chapters 2 and 3. I am grateful to the following institutions for their support: the
Austrian Science Foundation (FWF), the U.S. Army Research Office (through
MSI Cornell), the National Science Foundation, the Alfred P. Sloan Foundation,
and the Mittag-Leffler Institute (Stockholm).
Ithaca, June 1993 Bernd Sturmfels
Preface to the second edition
Computational Invariant Theory has seen a lot of progress since this book was
first published 14 years ago. Many new theorems have been proved, many new
algorithmshavebeendeveloped, andmanynewapplications havebeenexplored.
Among the numerous interesting research developments, particularly noteworthy
from our perspective are the methods developed by Gregor Kemper for finite
groups and by Harm Derksen on reductive groups. The relevant references in-
clude
Harm Derksen, Computation of reductive group invariants, Advances in Mathe-
matics 141, 366–384, 1999;
Gregor Kemper, Computing invariants of reductive groups in positive character-
istic, Transformation Groups 8, 159–176, 2003.
These two authors also co-authored the following excellent book which centers
around the questions raised in my chapters 2 and 4, but which goes much further
and deeper than what I had done:
HarmDerksenandGregorKemper,Computationalinvarianttheory(Encyclopae-
dia of mathematical sciences, vol.130), Springer, Berlin, 2002.
Inasense,thepresentneweditionof“AlgorithmsinInvariantTheory”maynow
serve the role of a first introductory text which can be read prior to the book
by Derksen and Kemper. In addition, I wish to recommend three other terrific
booksoninvarianttheorywhichdealwithcomputationalaspectsandapplications
outside of pure mathematics:
Karin Gatermann, Computer algebra methods for equivariant dynamical systems
(Lecture notes in mathematics, vol.1728), Springer, Berlin, 2000;
MaraNeusel,Invarianttheory,AmericanMathematicalSociety,Providence,R.I.,
2007;
Peter Olver, Classical invariant theory, Cambridge University Press, Cambridge,
1999.
Graduate students and researchers across the mathematical sciences will find it
worthwhile to consult these three books for further information on the beautiful
subject of classical invariant theory from a contempory perspective.
Berlin, January 2008 Bernd Sturmfels
Contents
1 Introduction 1
1.1 Symmetric polynomials 2
1.2 Gröbner bases 7
1.3 What is invariant theory? 14
1.4 Torus invariants and integer programming 19
2 Invariant theory of finite groups 25
2.1 Finiteness and degree bounds 25
2.2 Counting the number of invariants 29
2.3 The Cohen–Macaulay property 37
2.4 Reflection groups 44
2.5 Algorithms for computing fundamental invariants 50
2.6 Gröbner bases under finite group action 58
2.7 Abelian groups and permutation groups 64
3 Bracket algebra and projective geometry 77
3.1 The straightening algorithm 77
3.2 The first fundamental theorem 84
3.3 The Grassmann–Cayley algebra 94
3.4 Applications to projective geometry 100
3.5 Cayley factorization 110
3.6 Invariants and covariants of binary forms 117
3.7 Gordan’s finiteness theorem 129
4 Invariants of the general linear group 137
4.1 Representation theory of the general linear group 137
4.2 Binary forms revisited 147
4.3 Cayley’s (cid:2)-process and Hilbert finiteness theorem 155
4.4 Invariants and covariants of forms 161
4.5 Lie algebra action and the symbolic method 169
4.6 Hilbert’s algorithm 177
4.7 Degree bounds 185
References 191
Subject index 196
1
Introduction
Invariant theory is both a classical and a new area of mathematics. It played a
central role in 19th century algebra and geometry, yet many of its techniques
and algorithms were practically forgotten by the middle of the 20th century.
Withthefieldsofcombinatoricsandcomputersciencerevivingold-fashioned
algorithmic mathematics during the past twenty years, also classical invariant
theory has come to a renaissance. We quote from the expository article of Kung
and Rota (1984):
“Like the Arabian phoenix rising out of its ashes, the theory of invariants, pro-
nounced dead at the turn of the century, is once again at the forefront of mathe-
matics. During its long eclipse, the language of modern algebra was developed,
a sharp tool now at last being applied to the very purpose for which it was
invented.”
ThisquotereferstothefactthatthreeofHilbert’sfundamentalcontributions
tomodernalgebra,namely,theNullstellensatz,theBasisTheoremandtheSyzygy
Theorem, were first proved as lemmas in his invariant theory papers (Hilbert
1890, 1893). It is also noteworthy that, contrary to a common belief, Hilbert’s
main results in invariant theory yield an explicit finite algorithm for computing
afundamentalsetofinvariantsforallclassicalgroups. WewilldiscussHilbert’s
algorithm in Chap. 4.
Throughout this text we will take the complex numbers C to be our ground
field. The ring of polynomials f.x ;x ;:::;x / in n variables with complex
1 2 n
coefficients is denoted CŒx ;x ;:::;x (cid:3). All algorithms in this book will be
1 2 n
based upon arithmetic operations in the ground field only. This means that if
the scalars in our input data are contained in some subfield K (cid:2) C, then all
scalarsintheoutputalsolieinK.Suppose,forinstance,wespecifyanalgorithm
whose input is a finite set of n(cid:3)n-matricesover C,and whose output isa finite
subset of CŒx ;x ;:::;x (cid:3). We will usually apply such an algorithm to a set of
1 2 n
input matrices which have entries lying in the field Q of rational numbers. We
can then be sure that all output polynomials will lie in QŒx ;x ;:::;x (cid:3).
1 2 n
Chapter 1 starts out with a discussion of the ring of symmetric polynomials,
whichisthesimplestinstanceofaringofinvariants.InSect.1.2werecallsome
basics from the theory of Gröbner bases, and in Sect. 1.3 we give an elemen-
tary exposition of the fundamental problems in invariant theory. Section 1.4 is
independent and can be skipped upon first reading. It deals with invariants of
algebraic tori and their relation to integer programming. The results of Sect. 1.4
will be needed in Sect. 2.7 and in Chap. 4.
2 Introduction
1.1. Symmetric polynomials
Our starting point is the fundamental theorem on symmetric polynomials. This
is a basic result in algebra, and studying its proof will be useful to us in three
ways. First, we illustrate some fundamental questions in invariant theory with
their solution in the easiest case of the symmetric group. Secondly, the main
theorem on symmetric polynomials is a crucial lemma for several theorems to
follow, and finally, the algorithm underlying its proof teaches us some basic
computer algebra techniques.
A polynomial f 2 CŒx ;:::;x (cid:3) is said to be symmetric if it is invariant
1 n
under every permutation of the variables x ;x ;:::;x . For example, the poly-
1 2 n
nomialf WDx x Cx x isnotsymmetricbecausef .x ;x ;x /6Df .x ;x ;
1 1 2 1 3 1 1 2 3 1 2 1
x /Dx x Cx x .Ontheotherhand,f WDx x Cx x Cx x issymmetric.
3 1 2 2 3 2 1 2 1 3 2 3
Let ´ be a new variable, and consider the polynomial
g.´/D.´(cid:4)x /.´(cid:4)x /:::.´(cid:4)x /
1 2 n
D´n(cid:4)(cid:4) ´n(cid:2)1C(cid:4) ´n(cid:2)2(cid:4):::C.(cid:4)1/n(cid:4) :
1 2 n
We observe that the coefficients of g with respect to the new variable ´,
(cid:4) Dx Cx C:::Cx ;
1 1 2 n
(cid:4) Dx x Cx x C:::Cx x C:::Cx x ;
2 1 2 1 3 2 3 n(cid:2)1 n
(cid:4) Dx x x Cx x x C:::Cx x x ;
3 1 2 3 1 2 4 n(cid:2)2 n(cid:2)1 n
(cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5)
(cid:4) Dx x x (cid:5)(cid:5)(cid:5)x ;
n 1 2 3 n
aresymmetricintheoldvariablesx ;x ;:::;x .Thepolynomials(cid:4) ;(cid:4) ;:::;(cid:4)
1 2 n 1 2 n
2CŒx ;x ;:::;x (cid:3) are called the elementary symmetric polynomials.
1 2 n
Since the property to be symmetric is preserved under addition and multi-
plication of polynomials, the symmetric polynomials form a subring of CŒx ;
1
:::;x (cid:3). This implies that every polynomial expression p.(cid:4) ;(cid:4) ;:::;(cid:4) / in the
n 1 2 n
elementary symmetric polynomials is symmetric in CŒx ;:::;x (cid:3). For instance,
1 n
the monomial c (cid:5) (cid:4)(cid:2)1(cid:4)(cid:2)2:::(cid:4)(cid:2)n in the elementary symmetric polynomials is
1 2 n
symmetric and homogeneous of degree (cid:5) C2(cid:5) C:::Cn(cid:5) in the original
1 2 n
variables x ;x ;:::;x .
1 2 n
Theorem 1.1.1 (Main theorem on symmetric polynomials). Every symmetric
polynomial f in CŒx ;:::;x (cid:3) can be written uniquely as a polynomial
1 n
(cid:2) (cid:3)
f.x ;x ;:::;x /Dp (cid:4) .x ;:::;x /;:::;(cid:4) .x ;:::;x /
1 2 n 1 1 n n 1 n
in the elementary symmetric polynomials.
Proof. The proof to be presented here follows the one in van der Waerden
1.1. Symmetric polynomials 3
(1971). Let f 2 CŒx ;:::;x (cid:3) be any symmetric polynomial. Then the fol-
1 n
lowing algorithm rewrites f uniquely as a polynomial in (cid:4) ;:::;(cid:4) .
1 n
We sort the monomials in f using the degree lexicographic order, here de-
noted “(cid:6)”. In this order a monomial x˛1:::x˛n is smaller than another mono-
1 n P P
mial xˇ1:::xˇn if it has lower total degree (i.e., ˛ < ˇ ), or if they have
1 n i i
the same total degree and the first nonvanishing difference ˛ (cid:4)ˇ is negative.
i i
Foranymonomialx˛1:::x˛n occurringinthesymmetricpolynomialf also
1 n
all its images x˛1:::x˛n under any permutation (cid:4) of the variables occur in f.
(cid:3)1 (cid:3)n
Thisimpliesthatthe initialmonomialinit.f/Dc(cid:5)x(cid:4)1x(cid:4)2:::x(cid:4)n off satisfies
1 2 n
(cid:6) (cid:7)(cid:6) (cid:7) :::(cid:7) (cid:6) . By definition, the initial monomial is the largest monomial
1 2 n
with respect to the total order “(cid:6)” which appears with a nonzero coefficient
in f.
In our algorithm we now replace f by the new symmetric polynomial fQ WD
f (cid:4)c(cid:5)(cid:4)(cid:4)1(cid:2)(cid:4)2(cid:4)(cid:4)2(cid:2)(cid:4)3(cid:5)(cid:5)(cid:5)(cid:4)(cid:4)n(cid:2)1(cid:2)(cid:4)n(cid:4)(cid:4)n,westorethesummandc(cid:5)(cid:4)(cid:4)1(cid:2)(cid:4)2(cid:4)(cid:4)2(cid:2)(cid:4)3(cid:5)(cid:5)(cid:5)
1 2 n(cid:2)1 n 1 2
(cid:4)(cid:4)n(cid:2)1(cid:2)(cid:4)n(cid:4)(cid:4)n, and, if fQ is nonzero, then we return to the beginning of the pre-
n(cid:2)1 n
vious paragraph.
Why does this process terminate? By construction, the initial monomial of
c(cid:5)(cid:4)(cid:4)1(cid:2)(cid:4)2(cid:4)(cid:4)2(cid:2)(cid:4)3(cid:5)(cid:5)(cid:5)(cid:4)(cid:4)n(cid:2)1(cid:2)(cid:4)n(cid:4)(cid:4)n equalsinit.f/.Henceinthedifferencedefining
1 2 n(cid:2)1 n
fQ the two initial monomials cancel, and we get init.fQ/ (cid:6) init.f/. The set
of monomials m with m (cid:6) init.f/ is finite because their degree is bounded.
Hence the above rewriting algorithm must terminate because otherwise it would
generate an infinite decreasing chain of monomials.
It remains to be seen that the representation of symmetric polynomials in
terms of elementary symmetric polynomials is unique. In other words, we need
to show that the elementary symmetric polynomials(cid:4) ;:::;(cid:4) are algebraically
1 n
independent over C.
Suppose on the contrary that there is a nonzero polynomial p.y ;:::;y /
1 n
such that p.(cid:4) ;:::;(cid:4) / D 0 in CŒx ;:::;x (cid:3). Given any monomial y˛1(cid:5)(cid:5)(cid:5)y˛n
1 n 1 n 1 n
of p, we find that x˛1C˛2C:::C˛nx˛2C:::C˛n(cid:5)(cid:5)(cid:5)x˛n is the initial monomial of
1 2 n
(cid:4)˛1(cid:5)(cid:5)(cid:5)(cid:4)˛n. Since the linear map
1 n
.˛ ;˛ ;:::;˛ /7!.˛ C˛ C:::C˛ ; ˛ C:::C˛ ; :::; ˛ /
1 2 n 1 2 n 2 n n
is injective, all other monomials (cid:4)ˇ1:::(cid:4)ˇn in the expansion of p.(cid:4) ;:::;(cid:4) /
1 n 1 n
have a different initial monomial. The lexicographically largest monomial
x˛1C˛2C:::C˛nx˛2C:::C˛n(cid:5)(cid:5)(cid:5)x˛n is not cancelled by any other monomial, and
1 2 n
therefore p.(cid:4) ;:::;(cid:4) / 6D 0. This contradiction completes the proof of Theo-
1 n
rem 1.1.1. G
As an example for the above rewriting procedure, we write the bivariate
symmetric polynomial x3 C x3 as a polynomial in the elementary symmetric
1 2
polynomials:
x3Cx3 (cid:4)!(cid:4)3(cid:4)3x2x (cid:4)3x x2 (cid:4)!(cid:4)3(cid:4)3(cid:4) (cid:4) :
1 2 1 1 2 1 2 1 1 2