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An Introduction to Invariant Theory Alberto Daniel 2 Orebro¨ University Department of Science and Technology Matematik C, 76 – 90 ECTS

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An Introduction to Invariant Theory Alberto Daniel 2 Orebro¨ University Department of Science and Technology Matematik C, 76 – 90 ECTS Department of Science and Technology An Introduction to Invariant Theory Alberto Daniel 2 Orebro¨ University Department of Science and Technology Matematik C, 76 { 90 ECTS An Introduction to Invariant Theory Alberto Daniel January 5, 2017 Handledare: Holger Schellwat Examinator: Niklas Eriksen Sj¨alvst¨andigtarbete, 15 hp Matematik, C{niva, 76 { 90 hp 4 Acknowledgements I would like to thank the many people who have helped me with criticism and suggestions for this project, and in particular Holger Schellwat for reading, correcting, and commenting on the many preliminary drafts, as well as for many fruitful discussions. I would like to express my thanks to the Mathematics Department at the Orebro¨ University for its hospitality, generosity and teachings during the autumn semester 2016/2017. Specific thanks are due to the Linnaeus Palme programme for the wonderful opportunity they gave me to study in Sweden. I would like to express my thanks to the mathematics department at the Eduardo Mondlane university for many things they did for my studies. I also grateful to Sloane whose paper [Slo 77] provided the initial moti- vation for this project. 5 6 Abstract This work is an attempt to explain in some detail section III D of the pa- per: N.J.A. Sloane. "Error-correcting codes and Invariant Theory [Slo 77]: New application of a Nineteenth-Century Technique". For that, we will be concerned with polynomial invariants of finite groups which come from a group action. We will introduce the basic notions of Invariant Theory to give an almost self-contained proof of Molien's theorem, and also present applications on linear codes. 7 8 Contents 1 Invariant Theory 13 1.1 Preliminaries . 13 1.2 Getting started . 15 1.3 Ordering on the Monomials . 19 1.4 The division algorithm . 21 1.5 Monomial Ideals and Dickson's Lemma . 22 1.6 Symmetric polynomials . 24 1.7 Finite generation of invariants . 27 1.8 Molien's Theorem . 32 1.9 Linear codes . 35 9 10 Introduction The purpose of this project is to give an almost self contained introduction and clarify the proof of an amazing theorem of Molien, as presented in [Slo 77]. In this paper, we discuss invariant theory of finite groups. We begin by giving some preliminaries for invariants and we prove the fundamental theorem on symmetric functions. We shall prove the fundamental results of Hilbert and Noether for in- variant rings of finite linear groups. We will also derive the Molien's formula of the ring of invariants. We will show, through examples, that the Molien's formula helps us to see how many Invariants are linearly independent under finite group. Invariant Theory is concerned with the action of groups on rings and the invariants of the action. Here we will restrict ourselves to the left actions of finite linear groups on homogeneous polynomials rings with entries in the complex number C. In the 19th century it was found that the set of all homogeneous invariants under group G, that we will denote by RG, could be described fully by finite set of generators for several suggestive special cases of G. It soon became clear that the fundamental problem of Invariant Theory was to find necessary and sufficient conditions for RG to be finitely generated. In this paper we will give the answer of this problem. 11 12 Chapter 1 Invariant Theory 1.1 Preliminaries 1.1.1 Definition. The group G acts on the set X if for all g 2 G, there is a map G × X ! X; (g; x) 7! g:x such that 1. 8x 2 X; 8g; h 2 G : h:(g:x) = (hg):x. 2. 8x 2 X : 1:x = x. 1.1.2 Remark. An action of G on X may be viewed as a homomorphism G ! S(X) = fθ : θ : X ! X bijectiveg. Hence, we may write g(x) instead of g:x. We will use both notations. When X is a vector space we get the following. 1.1.3 Definition. A linear representation of a finite group G is a homo- morphism G −! L(V; V ). Here, L(V; V ) is the space of linear mappings from a vector space V to itself. The dimension of the vector space V is known as the dimension of the representation. Once we have chosen a basis for V the elements of L(V; V ) can be inter- preted as n×n matrices, where n is the dimension of the representation. The condition that the representation must be a homomorphism now becomes the condition 8g; h 2 G :[g][h] = [gh]: The multiplication on the left is matrix multiplication while the product on the right of this equation is multiplication in the group. This condition has the consequence that [g−1] = [g]−1; [id] = I: 13 1.1.4 Definition. Let V be a finite dimensional complex inner product space and T : V ! V linear. Then T is called unitary if 8u; v 2 V : hT (u);T (v)i = hu; vi: 1.1.5 Proposition. In this context, with [T ] 2 Mat(n; n; C) standard ma- trix w.r.t orthonormal basis. Then T is unitary () [T ]∗[T ] = I, where > [T ]∗ = [T ] () [T ] has orthonormal rows () [T ] has orthonormal columns. Proof. For vectors u; v of the same dimension, we have that u · v := u>v where the right-hand term is just matrix multiplication. n Let [T ] be defined as follows, with each vi 2 C , vi = [T (xi)]: [T ] = v1 j · · · j vn Now, note that [T ]> is equal to almost the same thing, except the columns are now the rows, and we have turned them up. it's equal to: 2 > 3 v1 > 6 . 7 [T ] = 4 . 5 > vn Now we just multiply by [T ], we obtain 2 > 3 2 > > 3 v1 v1 v1 ··· v1 vn > 6 . 7 6 . 7 [T ] [g] = 4 . 5 v1 j · · · j vn = 4 . 5 > > > vn vn v1 ··· v1 vn But, since u · v = u>v, we can simplify this nightmarish mess. 0 1 v1v1 ··· v1vn B . C = @ . A vnv1 ··· v1vn Since [T ] has orthonormal rows or columns, then vi · vj = 1 if i = j and vi · vj = 0 if i 6= j. Replacing these values in the matrix, we get: 0 1 0 ··· 0 1 B 0 1 ··· 0 C [T ]>[T ] = B C B . C @ . A 0 0 ··· 1 > Thus, [T ]>[T ] = I () [T ] [T ] = I. Now we start looking at the ring R = C[x1; ··· ; xn] of complex polyno- mials in n variables. 14 1.1.6 Definition. The polynomial f 2 R is homogeneous of degree d if f(tv) = tdf(v) for all v 2 V and t 2 C. Cleary, the product of polynomials f1 and f2 homogeneous of degrees d1 and d2 respectively, is again homogeneous polynomial of total degree d1 +d2 d d d +d since (f1f2)(tv) = f1(tv)f2(tv) = t 1 f1(v)t 2 f2(v) = t 1 2 (f1f2)(v). Now, it follows that every homogeneous polynomials f can be written as a sum X α f = aαx jαj=d n α α1 αn where α = (α1; ··· ; αn) 2 N , aα 2 C and x := x1 ··· xn . Every polyno- mial f over C can be expressed as a finite sum of homogeneous polynomials. The homogeneous polynomials that make up the polynomial f are called the homogeneous components of f. 1.1.7 Example. 1. f(x; y) = x2 + xy + y2 is a homogeneous polynomial of degree 2. 2. f(x) = x3 + 1 is not a homogeneous polynomial. 3. f(x; y; z) = x3 + xyz + yz2 + x2 + xy + yz + y + 2x + 6 is a polynomial that is the sum of four homogeneous polynomials: x3 + xyz + yz2 (of degree 3), x2 + xy + yz (of degree 2), y + 2x (of degree 1) and 6 (of degree 0). Let R be the algebra of polynomials in the variables x1; ··· ; xn with coefficients in C, i.e, R = C[x1; ··· ; xn]. Note that f 2 R is homogeneous if and only if all their components are the same total degree. In terms of the basis x = (x1; ··· ; xn), we have Rd = C[x1; ··· ; xn]d consisting of all homogeneous polynomials of degree d. 1.1.8 Definition. Rd := ff 2 R : f is homogeneous of degree dg. L 1.1.9 Proposition. R = d2N Rd. Since the monomials x1; ··· ; xn of degree one form a basis for R1, it follows that their products x2 := all monomials of total degree 2 form a d1 dn basis for R2, and, in general, the monomials x1 ··· xn for d1 + ··· + dn = d form a basis xd of Rd. 1.2 Getting started The ring R = C[x1; ··· ; xn] may be viewed as a complex vector space where multiplication by scalars is multiplication with constant polynomials. We want to make the connection between vector spaces and rings much clear by viewing the variables as linear forms. 15 From now on V denotes a complex inner product space with orthonormal standard basis e = (e1; ··· ; en) (only here ei denote standard basis vectors) and V ∗ = L(V; C) its algebraic dual space with orthonormal basis x = (x1; ··· ; xn) satisfying 81 ≤ i; j ≤ n : xi(ej) = δij. ∗ Thus, V = hx1; ··· ; xni and at the same time R = C[x], that is, the ∗ variables are linear forms. In particular, R1 = V (and R0 ≈ C) and R is an algebra. From now on we also fix a finite group G acting unitarily on V ∗, i.e. for every g 2 G, the mapping V ∗ ! V ∗; f 7! g:f (or f 7! g(f)) is unitary.
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