Polynomial Rings and Their Automorphisms

Home , Free algebra, Polynomial ring, Subring

Polynomial rings and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial Polynomial rings and their automorphisms ring Automorphisms and endomorphisms

The notions of Vipul Naik invariant subring Some questions about the invariant subring

More invariant April 23, 2007 subrings

Further connections

A summary Polynomial rings Outline and their automorphisms

A crash course in ring theory Vipul Naik Definition of ring Modules over rings Generating sets and bases A crash course in Rings and ideals ring theory Concept of subring Definition of ring Modules over rings The polynomial ring Generating sets and The polynomial ring in one variable bases The polynomial ring in many variables Rings and ideals Concept of subring Automorphisms and endomorphisms The polynomial Homomorphism of rings ring Homomorphisms from the polynomial ring Linear and affine endomorphisms Automorphisms and The notions of invariant subring endomorphisms The fixed-point relationship The notions of Some questions about the invariant subring invariant subring Representations and faithful representations Generating sets and questions Some questions about the invariant More invariant subrings subring The orthogonal group Relation between invariant polynomials and vanishing sets More invariant subrings Further connections The module of covariants Further Harmonic polynomials and the Laplacian connections

A summary A summary I (R, ∗) forms a semigroup (that is, ∗ is an associative binary operation)

I The following distributivity laws hold:

a ∗ (b + c) = (a ∗ b) + (a ∗ c) (a + b) ∗ c = (a ∗ c) + (b ∗ c)

The identity element for addition is denoted as 0 and the inverse operation is denoted by the preﬁx unary −.

Polynomial rings What’s a ring and their automorphisms

Vipul Naik

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary I The following distributivity laws hold:

a ∗ (b + c) = (a ∗ b) + (a ∗ c) (a + b) ∗ c = (a ∗ c) + (b ∗ c)

The identity element for addition is denoted as 0 and the inverse operation is denoted by the preﬁx unary −.

Polynomial rings What’s a ring and their automorphisms

Vipul Naik

A ring is a set R equipped with two binary operations + A crash course in ring theory (addition) and ∗ (multiplication) such that: Deﬁnition of ring Modules over rings Generating sets and I (R, +) forms an Abelian group bases Rings and ideals I (R, ∗) forms a semigroup (that is, ∗ is an associative Concept of subring The polynomial binary operation) ring

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings What’s a ring and their automorphisms

Vipul Naik

The notions of a ∗ (b + c) = (a ∗ b) + (a ∗ c) invariant subring Some questions about the invariant (a + b) ∗ c = (a ∗ c) + (b ∗ c) subring

More invariant The identity element for addition is denoted as 0 and the subrings inverse operation is denoted by the preﬁx unary −. Further connections

A summary In other words, the multiplication operation is a monoid operation. Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with 1 = 0 must be the trivial ring.

Polynomial rings Ring with identity and their automorphisms

Vipul Naik

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with 1 = 0 must be the trivial ring.

Polynomial rings Ring with identity and their automorphisms

Vipul Naik

A crash course in ring theory A ring with identity is a ring for which the multiplication Deﬁnition of ring Modules over rings Generating sets and operation has an identity element, that is, there exists an bases Rings and ideals element 1 ∈ R such that: Concept of subring The polynomial a ∗ 1 = 1 ∗ a = a ∀ a ∈ R ring Automorphisms and endomorphisms In other words, the multiplication operation is a monoid The notions of invariant subring operation. Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Ring with identity and their automorphisms

Vipul Naik

Further connections

A summary All the rings we shall be looking at today are so-called commutative rings with identity, viz ∗ is commutative and also has an identity element.

Polynomial rings Commutative ring and their automorphisms

Vipul Naik

A crash course in ring theory Deﬁnition of ring Modules over rings Generating sets and bases Rings and ideals A ring is said to be commutative if the multiplicative Concept of subring operation ∗ is commutative. The polynomial ring

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Commutative ring and their automorphisms

Vipul Naik

All the rings we shall be looking at today are so-called Automorphisms and commutative rings with identity, viz ∗ is commutative and endomorphisms

also has an identity element. The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary I We assume multiplication takes higher precedence over addition. This helps us leave out a number of parentheses. For instance, (a ∗ b) + (c ∗ d) can be written simply as ab + cd I Parentheses are also dropped from repeated addition I We denote by n ∈ N the number 1 + 1 + 1 ... 1 where we add 1 to itself n times. Moreover, we denote by nx the number x + x + ... + x (even when there doesn’t exist any 1) The first of these conventions is justified by associativity of multiplication, the second one is justified by the distributivity law.

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary I Parentheses are also dropped from repeated addition I We denote by n ∈ N the number 1 + 1 + 1 ... 1 where we add 1 to itself n times. Moreover, we denote by nx the number x + x + ... + x (even when there doesn’t exist any 1) The first of these conventions is justified by associativity of multiplication, the second one is justified by the distributivity law.

Polynomial rings Conventions followed while writing expressions in and their automorphisms a ring Vipul Naik We generally adopt the following conventions: A crash course in ring theory I The multiplication symbol, as well as parentheses for Deﬁnition of ring Modules over rings multiplication, are usually omitted. Thus, a ∗ (b ∗ c) Generating sets and bases may be simply written as abc Rings and ideals We assume multiplication takes higher precedence over Concept of subring I The polynomial addition. This helps us leave out a number of ring Automorphisms parentheses. For instance, (a ∗ b) + (c ∗ d) can be and written simply as ab + cd endomorphisms The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary I We denote by n ∈ N the number 1 + 1 + 1 ... 1 where we add 1 to itself n times. Moreover, we denote by nx the number x + x + ... + x (even when there doesn’t exist any 1) The first of these conventions is justified by associativity of multiplication, the second one is justified by the distributivity law.

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Conventions followed while writing expressions in and their automorphisms a ring Vipul Naik We generally adopt the following conventions: A crash course in ring theory I The multiplication symbol, as well as parentheses for Definition of ring Modules over rings multiplication, are usually omitted. Thus, a ∗ (b ∗ c) Generating sets and bases may be simply written as abc Rings and ideals We assume multiplication takes higher precedence over Concept of subring I The polynomial addition. This helps us leave out a number of ring Automorphisms parentheses. For instance, (a ∗ b) + (c ∗ d) can be and written simply as ab + cd endomorphisms The notions of I Parentheses are also dropped from repeated addition invariant subring I We denote by n ∈ N the number 1 + 1 + 1 ... 1 where Some questions about the invariant we add 1 to itself n times. Moreover, we denote by nx subring the number x + x + ... + x (even when there doesn’t More invariant subrings exist any 1) Further The first of these conventions is justified by associativity of connections multiplication, the second one is justified by the distributivity A summary law. Some examples of fields we have seen are Fp (the finite field on p elements), Q (the rationals), R (the reals) and C (the complex numbers). An example of a ring which is not a field is Z (the ring of integers). Another is Z/nZ (the ring of integers modulo n).

Polynomial rings Field and their automorphisms

Vipul Naik

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary An example of a ring which is not a ﬁeld is Z (the ring of integers). Another is Z/nZ (the ring of integers modulo n).

Polynomial rings Field and their automorphisms

Vipul Naik

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Field and their automorphisms

Vipul Naik

A crash course in ring theory Definition of ring Modules over rings Generating sets and A field is a very special kind of ring where the nonzero bases Rings and ideals elements form a group under multiplication. Concept of subring The polynomial Some examples of fields we have seen are Fp (the finite field ring on p elements), Q (the rationals), R (the reals) and C (the Automorphisms and complex numbers). endomorphisms An example of a ring which is not a field is Z (the ring of The notions of integers). Another is /n (the ring of integers modulo n). invariant subring Z Z Some questions about the invariant subring

More invariant subrings

Further connections

A summary I The map . deﬁnes a monoid action of the multiplicative monoid of k, over V (as Abelian group automorphisms). In simple language:

a.(v + w) = a.v + a.w a.(b.v) = (ab).v

I For any ﬁxed v ∈ V , the map k → V deﬁned by a 7→ a.v is a group homomorphism

Polynomial rings Vector space over a ﬁeld and their automorphisms

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Vector space over a ﬁeld and their automorphisms

Vipul Naik Let k be a field. A vector space over k is a set V equipped A crash course in with a binary operation + and an operation . : k × V → V ring theory Definition of ring such that: Modules over rings Generating sets and bases I (V , +) is an Abelian group. Rings and ideals Concept of subring I The map . defines a monoid action of the multiplicative The polynomial monoid of k, over V (as Abelian group ring Automorphisms automorphisms). In simple language: and endomorphisms

The notions of invariant subring

a.(v + w) = a.v + a.w Some questions about the invariant a.(b.v) = (ab).v subring More invariant subrings I For any ﬁxed v ∈ V , the map k → V deﬁned by Further a 7→ a.v is a group homomorphism connections

A summary I The map . deﬁnes a monoid action of the multiplicative monoid of R on M

Polynomial rings Module over a ring and their automorphisms

Vipul Naik

A crash course in ring theory Deﬁnition of ring Modules over rings Generating sets and bases Let R be a commutative ring with identity. A module over R Rings and ideals is a set M equipped with a binary operation and a map Concept of subring The polynomial . : R × M → M such that: ring

Automorphisms I (M, +) is an Abelian group. and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Module over a ring and their automorphisms

Vipul Naik

Automorphisms I (M, +) is an Abelian group. and endomorphisms The map . deﬁnes a monoid action of the multiplicative I The notions of monoid of R on M invariant subring Some questions about the invariant subring

More invariant subrings

Further connections

A summary In other words, a generating set is a subset such that every element is a R-linear combination of elements from the subset. A R-module that has a ﬁnite generating set is termed a ﬁnitely generated R-module.

Polynomial rings Generating set for a module and their automorphisms

Vipul Naik

A crash course in ring theory Deﬁnition of ring Modules over rings A set of elements m1, m2,..., mn is said to be a generating Generating sets and bases set for a R-module M if given any m ∈ M, we can express m Rings and ideals P Concept of subring as ri mi where ri ∈ R. The elements mi are termed i The polynomial generators. ring Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary A R-module that has a ﬁnite generating set is termed a ﬁnitely generated R-module.

Polynomial rings Generating set for a module and their automorphisms

Vipul Naik

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Generating set for a module and their automorphisms

Vipul Naik

A R-module that has a ﬁnite generating set is termed a Some questions about the invariant ﬁnitely generated R-module. subring

More invariant subrings

Further connections

A summary In other words, there are no unexpected dependencies between the generators. P In general, dependencies of the form i ri mi = 0 are termed relations, and a relation is trivial if and only if all the ri s are zero.

Polynomial rings Free generating set and their automorphisms

Vipul Naik

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary P In general, dependencies of the form i ri mi = 0 are termed relations, and a relation is trivial if and only if all the ri s are zero.

Polynomial rings Free generating set and their automorphisms

Vipul Naik

A crash course in ring theory A generating set m , m ,..., m for a R-module M is Deﬁnition of ring 1 2 n Modules over rings Generating sets and termed a free generating set if: bases Rings and ideals X Concept of subring ri mi = 0 =⇒ ri = 0 ∀ i The polynomial ring i Automorphisms and In other words, there are no unexpected dependencies endomorphisms between the generators. The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Free generating set and their automorphisms

Vipul Naik

zero. More invariant subrings

Further connections

A summary Clearly any free generating set is irredundant, because the ability to express one generator as a linear combination of the others deﬁnitely gives a relation.

Polynomial rings Irredundant generating set and their automorphisms

Vipul Naik

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Irredundant generating set and their automorphisms

Vipul Naik

A crash course in ring theory Definition of ring Modules over rings Generating sets and bases A generating set m1, m2,..., mn for a R-module M is Rings and ideals Concept of subring termed irredundant(defined) if no proper subset of it is a The polynomial generating set. ring Automorphisms Clearly any free generating set is irredundant, because the and ability to express one generator as a linear combination of endomorphisms The notions of the others definitely gives a relation. invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary The idea is to pick any generator with a nonzero coeﬃcient, say ri , and multiply the whole equation by 1/ri , and then transfer all the other terms to the right side. We can do this precisely because every nonzero element is invertible.

Polynomial rings In the case of ﬁelds and their automorphisms

Vipul Naik

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings In the case of ﬁelds and their automorphisms

Vipul Naik

A crash course in ring theory Definition of ring Modules over rings Generating sets and In the case of fields, the converse is also true. That is, any bases Rings and ideals irredundant generating set is free. In other words, given any Concept of subring nontrivial relation between the generators, we can express The polynomial one of the generators in terms of the others. ring Automorphisms The idea is to pick any generator with a nonzero coefficient, and endomorphisms say ri , and multiply the whole equation by 1/ri , and then The notions of transfer all the other terms to the right side. We can do this invariant subring precisely because every nonzero element is invertible. Some questions about the invariant subring

More invariant subrings

Further connections

A summary A module over a ring which possesses a free generating set is termed a free module(deﬁned).

Polynomial rings Vector spaces and free modules and their automorphisms

Vipul Naik

A crash course in ring theory Definition of ring Modules over rings Generating sets and bases For a field, every module (viz vector space) has an Rings and ideals Concept of subring irredundant generating set, which is also a free generating The polynomial set, and in the particular case of fields, we use the term ring Automorphisms basis(defined) for such a set. and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Vector spaces and free modules and their automorphisms

Vipul Naik

Some questions about the invariant subring

More invariant subrings

Further connections

A summary A submodule of a module is an additive subgroup that is closed under the ring action. An ideal(deﬁned) of a ring is a submodule of the ring when viewed naturally as a module over itself.

Polynomial rings Ring as a module over itself and their automorphisms

Vipul Naik

A crash course in ring theory Deﬁnition of ring Modules over rings Generating sets and Every ring is a module over itself. In fact, when we’re bases Rings and ideals dealing with a ring with identity, it is a free module over Concept of subring The polynomial itself with the generator being the element 1. ring

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary An ideal(deﬁned) of a ring is a submodule of the ring when viewed naturally as a module over itself.

Polynomial rings Ring as a module over itself and their automorphisms

Vipul Naik

A submodule of a module is an additive subgroup that is Automorphisms and closed under the ring action. endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Ring as a module over itself and their automorphisms

Vipul Naik

A submodule of a module is an additive subgroup that is Automorphisms and closed under the ring action. endomorphisms An ideal(deﬁned) of a ring is a submodule of the ring when The notions of viewed naturally as a module over itself. invariant subring Some questions about the invariant subring

More invariant subrings

Further connections

A summary In the particular case when the whole ring contains an identity element, we typically make the following added assumption about the subring: it contains the identity element of the whole ring.

Polynomial rings Deﬁnition of subring and their automorphisms

Vipul Naik

Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Deﬁnition of subring and their automorphisms

Vipul Naik

A crash course in ring theory Deﬁnition of ring Modules over rings Generating sets and bases A subset of a ring is said to be a subring if it is a ring with Rings and ideals Concept of subring the inherited addition and multiplication operations. The polynomial In the particular case when the whole ring contains an ring Automorphisms identity element, we typically make the following added and assumption about the subring: it contains the identity endomorphisms The notions of element of the whole ring. invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Outline and their automorphisms

The notions of invariant subring The notions of The ﬁxed-point relationship invariant subring

Some questions about the invariant subring Some questions Representations and faithful representations about the invariant Generating sets and questions subring

More invariant subrings More invariant The orthogonal group subrings Relation between invariant polynomials and vanishing sets Further Further connections connections The module of covariants A summary Harmonic polynomials and the Laplacian

A summary We are in particular interested in the polynomial ring in one variable over a ﬁeld.

Polynomial rings Deﬁnition of polynomial ring in one variable and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring The polynomial ring in one variable The polynomial ring Let R be a ring. The polynomial ring in one variable, or one in many variables indeterminate, is the set of all formal polynomials in one Automorphisms and variable, with addition and multiplication deﬁned as usual. endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Deﬁnition of polynomial ring in one variable and their automorphisms

Vipul Naik

A crash course in ring theory

We are in particular interested in the polynomial ring in one The notions of variable over a ﬁeld. invariant subring Some questions about the invariant subring

More invariant subrings

Further connections

A summary I The only invertible polynomials in one variable are the constant nonzero polynomials, viz the nonzero polynomials of degree zero

Polynomial rings Basic properties of the polynomial ring in one and their automorphisms variable Vipul Naik

A crash course in ring theory

The polynomial Here are some nice things about the polynomial ring in one ring The polynomial ring in one variable variable over a ﬁeld: The polynomial ring in many variables I If the product of two polynomials is the zero Automorphisms and polynomial, then one of the polynomials must be the endomorphisms zero polynomial. In other words, the product of two The notions of invariant subring nonzero polynomials must be a nonzero polynomial. Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Basic properties of the polynomial ring in one and their automorphisms variable Vipul Naik

A crash course in ring theory

constant nonzero polynomials, viz the nonzero More invariant polynomials of degree zero subrings Further connections

A summary From this, we can in fact deduce the fact that every polynomial over k[x] can be written uniquely as a product of irreducible polynomials, where an irreducible polynomial is one that cannot be factorized further.

Polynomial rings Ideals of the polynomial ring in one variable and their automorphisms

Vipul Naik

A crash course in ring theory

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Ideals of the polynomial ring in one variable and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring The polynomial ring If k is a ﬁeld, then any ideal in k[x] (viz, any in one variable The polynomial ring k[x]-submodule of k[x]) is a free module with 1 generator. in many variables In other words, it is what is called a principal ideal(deﬁned). Automorphisms and From this, we can in fact deduce the fact that every endomorphisms polynomial over k[x] can be written uniquely as a product of The notions of invariant subring

irreducible polynomials, where an irreducible polynomial is Some questions one that cannot be factorized further. about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Iterating the polynomial ring operation and their automorphisms

Vipul Naik

A crash course in For any ring R, we can consider the associated polynomial ring theory

ring R[x1]. Setting this as our new ring, we can consider the The polynomial ring next associated polynomial, R[x1][x2], which is basically The polynomial ring in one variable polynomials with x2 as the indeterminate, over the ring The polynomial ring R[x ]. We can do this repeatedly and get something called: in many variables 1 Automorphisms and endomorphisms

R[x1][x2] ... [xn] The notions of invariant subring

Now, because of the essentially commutative nature of Some questions about the invariant things, we can think of this as simply: subring

More invariant subrings R[x1, x2,..., xn] Further viz the polynomial ring in n variables connections A summary I The only invertible polynomials are the constant nonzero polynomials

I Every polynomial can be factorized uniquely as a product of irreducible polynomials (upto factors of multiplicative constants) It is not however true that every ideal is principal (we will not delve much into this).

Polynomial rings Nice properties that continue to hold for many and their automorphisms variables Vipul Naik

A crash course in ring theory

The polynomial Let k be a ﬁeld. Then the n-variate polynomial ring over k, ring The polynomial ring viz k[x1, x2,..., xn], satisﬁes the following: in one variable The polynomial ring in many variables I The product of any two nonzero polynomials is nonzero Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary I Every polynomial can be factorized uniquely as a product of irreducible polynomials (upto factors of multiplicative constants) It is not however true that every ideal is principal (we will not delve much into this).

Polynomial rings Nice properties that continue to hold for many and their automorphisms variables Vipul Naik

A crash course in ring theory

nonzero polynomials The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary It is not however true that every ideal is principal (we will not delve much into this).

Polynomial rings Nice properties that continue to hold for many and their automorphisms variables Vipul Naik

A crash course in ring theory

Further connections

A summary Polynomial rings Nice properties that continue to hold for many and their automorphisms variables Vipul Naik

A crash course in ring theory

nonzero polynomials The notions of invariant subring I Every polynomial can be factorized uniquely as a Some questions product of irreducible polynomials (upto factors of about the invariant subring multiplicative constants) More invariant subrings It is not however true that every ideal is principal (we will Further not delve much into this). connections A summary Polynomial rings Outline and their automorphisms

A crash course in ring theory Vipul Naik Definition of ring Modules over rings Generating sets and bases A crash course in Rings and ideals ring theory Concept of subring The polynomial The polynomial ring ring The polynomial ring in one variable The polynomial ring in many variables Automorphisms and Automorphisms and endomorphisms endomorphisms Homomorphism of rings Homomorphism of Homomorphisms from the polynomial ring rings Homomorphisms from Linear and affine endomorphisms the polynomial ring Linear and affine The notions of invariant subring endomorphisms The fixed-point relationship The notions of Some questions about the invariant subring invariant subring Representations and faithful representations Some questions Generating sets and questions about the invariant More invariant subrings subring The orthogonal group More invariant Relation between invariant polynomials and vanishing sets subrings

Further connections Further The module of covariants connections Harmonic polynomials and the Laplacian A summary A summary Polynomial rings Homomorphism of rings with identity and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial Let R and S be rings with identity. A homomorphism from ring Automorphisms R to S is a map f : R → S such that: and endomorphisms Homomorphism of rings Homomorphisms from the polynomial ring f (a + b) = f (a) + f (b) Linear and aﬃne endomorphisms f (ab) = f (a)f (b) The notions of invariant subring f (1) = 1 Some questions about the invariant subring

More invariant subrings

Further connections

A summary I An endomorphism(deﬁned) is a homomorphism from a ring to itself (need not be injective, surjective or bijective)

I An automorphism(deﬁned) is an isomorphism from a ring to itself, or equivalently, a bijective endomorphism of the ring

Polynomial rings Isomorphism, automorphism and endomorphism and their automorphisms

Vipul Naik

A crash course in ring theory

We deﬁne: The polynomial ring

I An isomorphism(deﬁned) is a bijective homomorphism of Automorphisms and rings endomorphisms Homomorphism of rings Homomorphisms from the polynomial ring Linear and aﬃne endomorphisms The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary I An automorphism(deﬁned) is an isomorphism from a ring to itself, or equivalently, a bijective endomorphism of the ring

Polynomial rings Isomorphism, automorphism and endomorphism and their automorphisms

Vipul Naik

A crash course in ring theory

We deﬁne: The polynomial ring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Isomorphism, automorphism and endomorphism and their automorphisms

Vipul Naik

A crash course in ring theory

We deﬁne: The polynomial ring

I An isomorphism(defined) is a bijective homomorphism of Automorphisms and rings endomorphisms Homomorphism of I An endomorphism(defined) is a homomorphism from a rings Homomorphisms from the polynomial ring ring to itself (need not be injective, surjective or Linear and affine bijective) endomorphisms The notions of invariant subring I An automorphism(defined) is an isomorphism from a ring Some questions to itself, or equivalently, a bijective endomorphism of about the invariant the ring subring More invariant subrings

Further connections

A summary In fact, for any choice of elements a1, a2,..., an ∈ R, there is a unique homomorphism from k[x1, x2,..., xn] to R which sends each xi to the corresponding ai .

Polynomial rings Suﬃces to locate images of indeterminates and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

Let k[x1, x2,..., xn] be a polynomial ring, and R be another Automorphisms and ring containing a copy of k. Then, the injective endomorphisms Homomorphism of homomorphism from k to R can be extended to a rings Homomorphisms from homomorphism from k[x1, x2,..., xn] to R in many ways. the polynomial ring Linear and aﬃne endomorphisms The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Suﬃces to locate images of indeterminates and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

sends each xi to the corresponding ai . Some questions about the invariant subring

More invariant subrings

Further connections

A summary Here, the ring R is k[x1, x2,..., xn] itself. Hence, to specify the endomorphism, we need to give polynomials p1, p2,..., pn (each being a polynomial in all the xi s) such that each xi maps to the corresponding pi . Thus, every endomorphism of the polynomial ring (that ﬁxes the base ﬁeld pointwise) can be described by an arbitrary sequence of n polynomials in the indeterminates.

Polynomial rings Endomorphisms of the polynomial ring and their automorphisms

Vipul Naik

A crash course in ring theory Let’s now consider the problem of describing all The polynomial ring homomorphisms from the polynomial ring k[x1, x2,..., xn] to Automorphisms itself, which restrict to the identity on k. and endomorphisms Homomorphism of rings Homomorphisms from the polynomial ring Linear and aﬃne endomorphisms The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Thus, every endomorphism of the polynomial ring (that ﬁxes the base ﬁeld pointwise) can be described by an arbitrary sequence of n polynomials in the indeterminates.

Polynomial rings Endomorphisms of the polynomial ring and their automorphisms

Vipul Naik

A crash course in ring theory Let’s now consider the problem of describing all The polynomial ring homomorphisms from the polynomial ring k[x1, x2,..., xn] to Automorphisms itself, which restrict to the identity on k. and endomorphisms Here, the ring R is k[x1, x2,..., xn] itself. Hence, to specify Homomorphism of rings the endomorphism, we need to give polynomials Homomorphisms from the polynomial ring Linear and aﬃne p1, p2,..., pn (each being a polynomial in all the xi s) such endomorphisms that each xi maps to the corresponding pi . The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Endomorphisms of the polynomial ring and their automorphisms

Vipul Naik

A crash course in ring theory Let’s now consider the problem of describing all The polynomial ring homomorphisms from the polynomial ring k[x1, x2,..., xn] to Automorphisms itself, which restrict to the identity on k. and endomorphisms Here, the ring R is k[x1, x2,..., xn] itself. Hence, to specify Homomorphism of rings the endomorphism, we need to give polynomials Homomorphisms from the polynomial ring Linear and affine p1, p2,..., pn (each being a polynomial in all the xi s) such endomorphisms that each xi maps to the corresponding pi . The notions of invariant subring Thus, every endomorphism of the polynomial ring (that fixes Some questions the base field pointwise) can be described by an arbitrary about the invariant subring sequence of n polynomials in the indeterminates. More invariant subrings

Further connections

A summary In particular an endomorphism p is invertible if we can ﬁnd a q such that (q ◦ p)(xi ) = xi for each i.

Polynomial rings Composing two endomorphisms and their automorphisms

Vipul Naik

A crash course in ring theory

The rule for composing endomorphisms of the polynomial The polynomial ring is as follows: If the endomorphisms are ring Automorphisms p = (p1, p2,..., pn) and q = (q1, q2,..., qn) then their and composite q ◦ p is the endomorphism endomorphisms Homomorphism of rings Homomorphisms from the polynomial ring Linear and aﬃne endomorphisms xi 7→ qi (p1(x1, x2,..., xn), p2(x1, x2,..., xn),..., pn(x1, x2,..., xn)) The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Composing two endomorphisms and their automorphisms

Vipul Naik

A crash course in ring theory

Some questions about the invariant In particular an endomorphism p is invertible if we can ﬁnd a subring q such that (q ◦ p)(xi ) = xi for each i. More invariant subrings

Further connections

A summary It’s clear that the only polynomials which have an inverse in the composition sense are the linear polynomials. In other words, the automorphism group of the polynomial ring in one variable is the group of aﬃne maps x 7→ ax + b.

Polynomial rings Endomorphisms of the polynomial ring in one and their automorphisms variable Vipul Naik

A crash course in ring theory

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Endomorphisms of the polynomial ring in one and their automorphisms variable Vipul Naik

A crash course in ring theory

The polynomial ring In the case of k[x] (polynomial ring in one variable), the Automorphisms and endomorphisms are described simply by polynomials. endomorphisms Homomorphism of Composition of endomorphisms is, in this context, rings Homomorphisms from the polynomial ring composition of polynomials. Linear and aﬃne It’s clear that the only polynomials which have an inverse in endomorphisms The notions of the composition sense are the linear polynomials. In other invariant subring words, the automorphism group of the polynomial ring in Some questions about the invariant one variable is the group of aﬃne maps x 7→ ax + b. subring More invariant subrings

Further connections

A summary More speciﬁcally, if we consider endomorphisms where all the pi s are homogeneous linear polynomials (viz, linear polynomials without a constant term), we get something which corresponds to linear maps on the vector space kn (basis vectors viewed as xi s) Among these, the invertible elements are precisely those which correspond to invertible aﬃne (respectively linear) maps – viz GAn(k) (respectively GLn(k)).

Polynomial rings Aﬃne endomorphisms in more than one variable and their automorphisms

Vipul Naik

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Among these, the invertible elements are precisely those which correspond to invertible aﬃne (respectively linear) maps – viz GAn(k) (respectively GLn(k)).

Polynomial rings Aﬃne endomorphisms in more than one variable and their automorphisms

Vipul Naik

A crash course in Given k[x1, x2,..., xn] we can consider endomorphisms ring theory where all the p s are linear polynomials in the x s. This The polynomial i i ring n corresponds to affine maps on the vector space k (basis Automorphisms and vectors viewed as xi s) endomorphisms Homomorphism of More specifically, if we consider endomorphisms where all the rings Homomorphisms from pi s are homogeneous linear polynomials (viz, linear the polynomial ring Linear and affine polynomials without a constant term), we get something endomorphisms which corresponds to linear maps on the vector space kn The notions of invariant subring

(basis vectors viewed as xi s) Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings Aﬃne endomorphisms in more than one variable and their automorphisms

Vipul Naik

(basis vectors viewed as xi s) Some questions about the invariant Among these, the invertible elements are precisely those subring

which correspond to invertible aﬃne (respectively linear) More invariant subrings maps – viz GAn(k) (respectively GLn(k)). Further connections

A summary There’s something nice about the polynomial automorphisms that come from linear automorphisms. Namely, these automorphisms actually preserve the degree of the polynomial. Any automorphism that is not linear will not preserve the degree of at least some polynomial.

Polynomial rings The upshot and their automorphisms

Vipul Naik

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings The upshot and their automorphisms

Vipul Naik

A crash course in The upshot is that: ring theory The polynomial ring GLn(k) ≤ GAn(k) ≤ Aut(k[x1, x2,..., xn]) Automorphisms and endomorphisms In other words, every linear automorphism (more generally Homomorphism of rings every aﬃne automorphism) gives rise to a polynomial Homomorphisms from the polynomial ring Linear and aﬃne automorphism, and this association is faithful. endomorphisms There’s something nice about the polynomial automorphisms The notions of that come from linear automorphisms. Namely, these invariant subring Some questions automorphisms actually preserve the degree of the about the invariant polynomial. Any automorphism that is not linear will not subring More invariant preserve the degree of at least some polynomial. subrings Further connections

A summary Polynomial rings Outline and their automorphisms

A crash course in ring theory Vipul Naik Definition of ring Modules over rings Generating sets and bases A crash course in Rings and ideals ring theory Concept of subring The polynomial The polynomial ring ring The polynomial ring in one variable The polynomial ring in many variables Automorphisms and Automorphisms and endomorphisms endomorphisms Homomorphism of rings The notions of Homomorphisms from the polynomial ring invariant subring Linear and affine endomorphisms The fixed-point relationship The notions of invariant subring The fixed-point relationship Some questions about the invariant Some questions about the invariant subring subring Representations and faithful representations Generating sets and questions More invariant subrings More invariant subrings The orthogonal group Further Relation between invariant polynomials and vanishing sets connections

Further connections A summary The module of covariants Harmonic polynomials and the Laplacian

A summary Thus, given any set P of polynomials p, we can consider the set of all automorphisms σ for which every p ∈ P is a ﬁxed point. This set of automorphisms is clearly a subgroup of the automorphism group. Analogously, for every subset S of the automorphism group, we can consider the set Q of all polynomials that are ﬁxed points of every σ ∈ S. Note that this set Q is clearly a ring.

Polynomial rings The ﬁxed-point relationship and their automorphisms

Vipul Naik

The notions of invariant subring The ﬁxed-point relationship Some questions about the invariant subring

More invariant subrings

Further connections

A summary Analogously, for every subset S of the automorphism group, we can consider the set Q of all polynomials that are ﬁxed points of every σ ∈ S. Note that this set Q is clearly a ring.

Polynomial rings The ﬁxed-point relationship and their automorphisms

Vipul Naik

A crash course in ring theory Given an automorphism σ of the polynomial ring, and a The polynomial polynomial p that sits inside this polynomial ring, we say ring Automorphisms that p is a fixed point of σ if σ(p) = p. and endomorphisms Thus, given any set P of polynomials p, we can consider the The notions of set of all automorphisms σ for which every p ∈ P is a fixed invariant subring The fixed-point point. This set of automorphisms is clearly a subgroup of the relationship Some questions automorphism group. about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings The ﬁxed-point relationship and their automorphisms

Vipul Naik

points of every σ ∈ S. Note that this set Q is clearly a ring. Further connections

A summary Polynomial rings Notion of Galois correspondence and their automorphisms

Vipul Naik

The above can be ﬁtted into the framework of a Galois A crash course in correspondence. ring theory The polynomial Given two sets A and B and a relation R between A and B, ring

the Galois correspondence for R is a pair of maps Automorphisms A B B A and S : 2 → 2 and T : 2 → 2 deﬁned as: endomorphisms

The notions of I For C ≤ A, S(C) is the set of all elements in B that are invariant subring The ﬁxed-point related to every element in C relationship Some questions I For D ≤ B, T (D) is the set of all elements in A that about the invariant are related to every element in D subring More invariant Then we have: subrings Further connections I C1 ⊆ C2 =⇒ S(C2) ⊆ S(C1) and similarly for T A summary I S ◦ T ◦ S = S and T ◦ S ◦ T = T Polynomial rings How this ﬁts in and their automorphisms

Vipul Naik

A crash course in ring theory

In our case, the relation is the fixed-point relation. That is, The polynomial the two sets are: ring Automorphisms and I A is the set of all polynomials endomorphisms B is the group GL(V ) The notions of I invariant subring The fixed-point I R is the relation of the given polynomial relationship Some questions We are interested in taking subgroups of GL(V ) and asking about the invariant for the invariant subrings, or conversely, in taking subrings of subring More invariant A and asking for the fixing subgroups. This essentially subrings corresponds to computing the maps T and S. Further connections

A summary Polynomial rings Outline and their automorphisms

A crash course in ring theory Vipul Naik Definition of ring Modules over rings Generating sets and bases A crash course in Rings and ideals ring theory Concept of subring The polynomial The polynomial ring ring The polynomial ring in one variable The polynomial ring in many variables Automorphisms and Automorphisms and endomorphisms endomorphisms Homomorphism of rings The notions of Homomorphisms from the polynomial ring invariant subring Linear and affine endomorphisms Some questions The notions of invariant subring about the invariant The fixed-point relationship subring Representations and Some questions about the invariant subring faithful Representations and faithful representations representations Generating sets and questions Generating sets and questions More invariant subrings More invariant The orthogonal group subrings Relation between invariant polynomials and vanishing sets Further Further connections connections The module of covariants Harmonic polynomials and the Laplacian A summary

A summary The representation is said to be faithful(deﬁned) if ρ is an injective map. In other words, we view G as a subgroup of GL(V ).

Polynomial rings Representation of a group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring (deﬁned) Let G be a group. A linear representation of G over Automorphisms and a ﬁeld k is a homomorphism ρ : G → GL(V ) where V is a endomorphisms

vector space over k and GL(V ) is the group of k-linear The notions of automorphisms of V . invariant subring Some questions about the invariant subring Representations and faithful representations Generating sets and questions More invariant subrings

Further connections

A summary Polynomial rings Representation of a group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring (deﬁned) Let G be a group. A linear representation of G over Automorphisms and a ﬁeld k is a homomorphism ρ : G → GL(V ) where V is a endomorphisms

vector space over k and GL(V ) is the group of k-linear The notions of automorphisms of V . invariant subring Some questions The representation is said to be faithful(deﬁned) if ρ is an about the invariant injective map. In other words, we view G as a subgroup of subring Representations and faithful GL(V ). representations Generating sets and questions More invariant subrings

Further connections

A summary This is in essence the same as the question we considered earlier.

Polynomial rings Invariant subring for a representation and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

One of the many aspects to a representation of a group is Automorphisms and the following: What is the subring of polynomials that are endomorphisms

invariant under the action of the group? In other words, The notions of what are the polynomials that are unchanged under the invariant subring Some questions action of the group on the xi s? about the invariant subring Representations and faithful representations Generating sets and questions More invariant subrings

Further connections

A summary Polynomial rings Invariant subring for a representation and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

One of the many aspects to a representation of a group is Automorphisms and the following: What is the subring of polynomials that are endomorphisms

invariant under the action of the group? In other words, The notions of what are the polynomials that are unchanged under the invariant subring Some questions action of the group on the xi s? about the invariant subring This is in essence the same as the question we considered Representations and faithful earlier. representations Generating sets and questions More invariant subrings

Further connections

A summary This is precisely the same as the ring of invariant polynomials corresponding to the symmetric group embedded naturally as permutations of the basis elements, in GLn(k).

Polynomial rings Symmetric polynomials and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring (deﬁned) A polynomial is said to be a symmetric polynomial if Automorphisms and it remains unchanged under any permutation of the xi s. endomorphisms

Clearly, the symmetric polynomials form a subring of the ring The notions of of all polynomials. invariant subring Some questions about the invariant subring Representations and faithful representations Generating sets and questions More invariant subrings

Further connections

A summary Polynomial rings Symmetric polynomials and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring (deﬁned) A polynomial is said to be a symmetric polynomial if Automorphisms and it remains unchanged under any permutation of the xi s. endomorphisms

Clearly, the symmetric polynomials form a subring of the ring The notions of of all polynomials. invariant subring Some questions This is precisely the same as the ring of invariant polynomials about the invariant corresponding to the symmetric group embedded naturally subring Representations and faithful as permutations of the basis elements, in GLn(k). representations Generating sets and questions More invariant subrings

Further connections

A summary We shall use the letter sj to denote the elementary symmetric polynomial of degree j.

Polynomial rings Elementary symmetric polynomials and their automorphisms

Vipul Naik

A crash course in ring theory

The elementary symmetric polynomial of degree j over The polynomial n−j ring variables x1, x2,..., xn is defined as the coefficient of x in Automorphisms the expression: and endomorphisms Y The notions of (x + xi ) invariant subring i Some questions about the invariant Or equivalently, as (−1)j times the coefficient of xn−j in the subring Representations and Q faithful expression i (x − xi ). representations Generating sets and questions More invariant subrings

Further connections

A summary Polynomial rings Elementary symmetric polynomials and their automorphisms

Vipul Naik

A crash course in ring theory

Further connections

A summary I This mapping is surjective. That is, any symmetric polynomial in the xi s can be expressed as a polynomial in the sj s.

Polynomial rings Two remarkable facts and their automorphisms

Vipul Naik It is clear that any elementary symmetric polynomial is a symmetric polynomial. Thus, any polynomial in terms of the A crash course in ring theory

elementary symmetric polynomials also is an elementary The polynomial symmetric polynomial. In other words, we have a ring Automorphisms homomorphism: and endomorphisms

Sn The notions of k[s1, s2,..., sn] → k[x1, x2,..., xn] invariant subring

Further connections

A summary Polynomial rings Two remarkable facts and their automorphisms

Vipul Naik It is clear that any elementary symmetric polynomial is a symmetric polynomial. Thus, any polynomial in terms of the A crash course in ring theory

elementary symmetric polynomials also is an elementary The polynomial symmetric polynomial. In other words, we have a ring Automorphisms homomorphism: and endomorphisms

Sn The notions of k[s1, s2,..., sn] → k[x1, x2,..., xn] invariant subring

Some questions Two remarkable facts are: about the invariant subring Representations and I This mapping is injective. That is, any two different faithful representations polynomials in the sj s give rise to different polynomials Generating sets and questions in the xi s. More invariant subrings I This mapping is surjective. That is, any symmetric Further polynomial in the xi s can be expressed as a polynomial connections in the sj s. A summary This gives some notions. Let k be a base field. Then any ring R containing k is termed a k-algebra. A generating set for R is a set S such that every element of R can be expressed as a polynomial in elements of S with coefficients from k.

Polynomial rings Generating set for an algebra and their automorphisms

Vipul Naik

Further connections

A summary Polynomial rings Generating set for an algebra and their automorphisms

Vipul Naik

A crash course in ring theory What we have done is shown that the invariant subring for The polynomial ring the symmetric group is in fact itself isomorphic to a Automorphisms polynomial ring, in other words, we can find polynomials in and endomorphisms it such that this subring is generated by these polynomials, The notions of without any further relations between them. invariant subring Some questions This gives some notions. Let k be a base field. Then any about the invariant ring R containing k is termed a k-algebra. A generating set subring Representations and faithful for R is a set S such that every element of R can be representations Generating sets and expressed as a polynomial in elements of S with coefficients questions from k. More invariant subrings

Further connections

A summary I An algebra over k is said to be free(deﬁned) if we can ﬁnd a generating set such that the mapping from the polynomial ring of that generating set to the given algebra, is an isomorphism.

Polynomial rings Generating set (continued) and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring I An algebra over k is said to be finitely generated(defined) Automorphisms if it has a finite generating set as a k-algebra, that is, and there is a surjective homomorphism to it from the endomorphisms The notions of polynomial ring in finitely many variables invariant subring

Some questions about the invariant subring Representations and faithful representations Generating sets and questions More invariant subrings

Further connections

A summary Polynomial rings Generating set (continued) and their automorphisms

Vipul Naik

A crash course in ring theory

An algebra over k is said to be free(deﬁned) if we can ﬁnd Some questions I about the invariant a generating set such that the mapping from the subring Representations and faithful polynomial ring of that generating set to the given representations Generating sets and algebra, is an isomorphism. questions More invariant subrings

Further connections

A summary I Is R a free k-algebra? That is, can R be viewed as the polynomial ring in some number of variables? In the case where G is the symmetric group, the answer to both questions was yes, the elementary symmetric polynomials formed a ﬁnite freely generating set for R.

Polynomial rings Two questions of interest and their automorphisms

Vipul Naik

Further connections

A summary Polynomial rings Two questions of interest and their automorphisms

Vipul Naik

A crash course in Given a group G and a (without loss of generality, faithful) ring theory The polynomial linear representation of G of degree n, let ring G R = k[x1, x2,..., xn] be the invariant subring Automorphisms and corresponding to G. Two questions we are interested in are: endomorphisms The notions of I Is R a ﬁnitely generated k-algebra? invariant subring Some questions I Is R a free k-algebra? That is, can R be viewed as the about the invariant subring polynomial ring in some number of variables? Representations and faithful representations In the case where G is the symmetric group, the answer to Generating sets and questions both questions was yes, the elementary symmetric More invariant polynomials formed a ﬁnite freely generating set for R. subrings Further connections

A summary Polynomial rings Outline and their automorphisms

Some questions about the invariant subring More invariant Representations and faithful representations subrings Generating sets and questions The orthogonal group Relation between More invariant subrings invariant polynomials The orthogonal group and vanishing sets Relation between invariant polynomials and vanishing sets Further Further connections connections The module of covariants A summary Harmonic polynomials and the Laplacian

A summary Equivalently, it is the group of those transformation of the space kn that ﬁx the origin and preserve the norm of any P 2 vector, that is, they preserve i xi for any vector (x1, x2,..., xn). Equivalently, it is the group of those transformations of the space kn that preserve the scalar product of any two vectors.

Polynomial rings Deﬁnition of the orthogonal group and their automorphisms

Vipul Naik

A crash course in ring theory

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary Polynomial rings Deﬁnition of the orthogonal group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial The orthogonal group of order n over a field k, denoted as ring On(k), is defined as the group of those matrices A such that Automorphisms T and AA is the identity matrix. endomorphisms Equivalently, it is the group of those transformation of the The notions of space kn that fix the origin and preserve the norm of any invariant subring 2 Some questions vector, that is, they preserve P x for any vector about the invariant i i subring (x1, x2,..., xn). More invariant Equivalently, it is the group of those transformations of the subrings n The orthogonal group space k that preserve the scalar product of any two vectors. Relation between invariant polynomials and vanishing sets Further connections

A summary It turns out that the converse is also true: any polynomial in the xi s that is invariant under the action of the orthogonal P 2 group must actually be a polynomial in i xi .

Polynomial rings Invariant polynomials for the orthogonal group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring P 2 Clearly, the polynomial i xi is an invariant polynomial Automorphisms under the action of the orthogonal group. Hence, the and endomorphisms

invariant subring contains, as a subring, the polynomial ring The notions of P 2 invariant subring generated by i xi . Some questions about the invariant subring

More invariant subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary Polynomial rings Invariant polynomials for the orthogonal group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring P 2 Clearly, the polynomial i xi is an invariant polynomial Automorphisms under the action of the orthogonal group. Hence, the and endomorphisms

invariant subring contains, as a subring, the polynomial ring The notions of P 2 invariant subring generated by i xi . Some questions It turns out that the converse is also true: any polynomial in about the invariant subring the xi s that is invariant under the action of the orthogonal P 2 More invariant group must actually be a polynomial in i xi . subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary It turns out that the action is also transitive, i.e. given any two points on the same sphere, there is an element of the orthogonal group taking one to the other. This essentially follows from the fact that any unit vector can be completed to an orthonormal basis.

Polynomial rings A closer inspection of the orthogonal group and their automorphisms action Vipul Naik

A crash course in ring theory

More invariant subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary Polynomial rings A closer inspection of the orthogonal group and their automorphisms action Vipul Naik

A crash course in ring theory

The polynomial The orthogonal group acts on the space kn, and kn naturally ring Automorphisms decomposes into orbits under the action. Since every and element of the orthogonal group preserves the polynomial endomorphisms P 2 The notions of i xi , each orbit must lie inside a “sphere” of the form invariant subring P 2 i xi = c for some value of c. Some questions about the invariant It turns out that the action is also transitive, i.e. given any subring two points on the same sphere, there is an element of the More invariant subrings orthogonal group taking one to the other. This essentially The orthogonal group Relation between invariant polynomials follows from the fact that any unit vector can be completed and vanishing sets to an orthonormal basis. Further connections

A summary We’ll prove a more general statement: Let p be a polynomial in x1, x2,..., xn. Any polynomial f such that f is constant on each locus p(x) = c (i.e. p(x) = p(y) =⇒ f (x) = f (y)) must itself be a polynomial in p.

Polynomial rings The proof for the invariant subring and their automorphisms

Vipul Naik

A crash course in ring theory

Some questions about the invariant subring

More invariant subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary Polynomial rings The proof for the invariant subring and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial Proving that the invariant subring comprises polynomials in ring P 2 i xi thus reduces to proving that: Automorphisms and Any polynomial that is constant on spheres (that is, loci of endomorphisms P 2 P 2 the form i xi = c) must be a polynomial in i xi . The notions of We’ll prove a more general statement: invariant subring Some questions Let p be a polynomial in x1, x2,..., xn. Any polynomial f about the invariant subring such that f is constant on each locus p(x) = c (i.e. More invariant p(x) = p(y) =⇒ f (x) = f (y)) must itself be a polynomial subrings The orthogonal group in p. Relation between invariant polynomials and vanishing sets Further connections

A summary Now, h also satisﬁes the property of being constant on every locus p(x) = c0. By induction, we can write h as a polynomial in p, and hence f can also be written as a polynomial in p.

Polynomial rings Proof of the general statement and their automorphisms

Vipul Naik

A crash course in ring theory

We write x for the tuple (x1, x2,..., xn). The polynomial Consider the locus p(x) = c. Suppose f (x) takes the value ring Automorphisms λ on this locus. Then, by the factor theorem: and endomorphisms

The notions of f (x) − λ = h(x)(p(x) − c) invariant subring

Some questions where h(x) is another polynomial. about the invariant subring

More invariant subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary Polynomial rings Proof of the general statement and their automorphisms

Vipul Naik

A crash course in ring theory

The notions of f (x) − λ = h(x)(p(x) − c) invariant subring

Some questions where h(x) is another polynomial. about the invariant Now, h also satisﬁes the property of being constant on every subring 0 More invariant locus p(x) = c . By induction, we can write h as a subrings The orthogonal group polynomial in p, and hence f can also be written as a Relation between invariant polynomials polynomial in p. and vanishing sets Further connections

A summary Polynomial rings Upshot: for the orthogonal group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

Automorphisms and We have shown that for the orthogonal group, the invariant endomorphisms The notions of subring is in fact the polynomial ring in one variable. Hence, invariant subring

it is both free and ﬁnitely generated. Some questions about the invariant subring

More invariant subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary Polynomial rings Orbits as sets of constancy and their automorphisms

Vipul Naik

A crash course in If G ≤ GL(V ), then any orbit of kn under the action of G, ring theory The polynomial must take a constant value under any polynomial invariant ring under the action of G. In other words, we can deﬁne two Automorphisms and relations: endomorphisms n The notions of I Given a subring R of the polynomial ring, call x, y ∈ k invariant subring as R-equivalent if f (x) = f (y) for any f ∈ R Some questions about the invariant n subring I Given a group G ≤ GL(V ), call x, y ∈ k as More invariant G-equivalent if there exists g ∈ G such that g.x = y subrings The orthogonal group Relation between Then if R is the invariant subring for G, G-equivalence invariant polynomials and vanishing sets implies R-equivalence. Further connections

A summary Polynomial rings Rings of constant functions versus ideals and their automorphisms

Vipul Naik

A crash course in ring theory

n The polynomial Given a subset S ⊆ k , deﬁne I (S) as the set of all ring polynomials that vanish at every point of S, and R(S) as the Automorphisms and ring of all polynomials that are constant on S. Then: endomorphisms The notions of invariant subring

R(S) = I (S) + k Some questions about the invariant In other words, every polynomial constant on S can be subring More invariant written as a polynomial that vanishes on S, plus a constant subrings polynomials. The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary I Hence, it is the intersection, over each orbit O of G, of the ring of polynomials constant on O, viz R(O):

G \ k[x1, x2,..., xn] = R(O) O

I Since R(O) = I (O) + k, we get:

G \ k[x1, x2,..., xn] = I (O) + k O

Polynomial rings Expression for the invariant subring and their automorphisms

Vipul Naik

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings The orthogonal group Relation between invariant polynomials and vanishing sets Further connections

A summary Polynomial rings Expression for the invariant subring and their automorphisms

Vipul Naik

Here’s the chain of reasoning: A crash course in ring theory Any polynomial invariant under the action of G must be The polynomial I ring constant on all the G-orbits Automorphisms and I Hence, it is the intersection, over each orbit O of G, of endomorphisms the ring of polynomials constant on O, viz R(O): The notions of invariant subring G \ Some questions k[x1, x2,..., xn] = R(O) about the invariant subring O More invariant Since R(O) = I (O) + k, we get: subrings I The orthogonal group Relation between invariant polynomials G \ and vanishing sets k[x1, x2,..., xn] = I (O) + k Further O connections A summary Polynomial rings Outline and their automorphisms

Some questions about the invariant subring More invariant Representations and faithful representations subrings Generating sets and questions Further More invariant subrings connections The orthogonal group The module of Relation between invariant polynomials and vanishing sets covariants Harmonic polynomials Further connections and the Laplacian The module of covariants A summary Harmonic polynomials and the Laplacian

A summary I A group G acting on GL(V ) and hence acting as algebra automorphisms of k[x1, x2,..., xn] G I R = A is the subring comprising invariant polynomials Since A is a ring containing R, A is a R-algebra, and in particular, A is also a R-module. A, viewed as a R-module, is termed the module of covariants(deﬁned).

Polynomial rings The module of covariants and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial The setup so far is: ring

Automorphisms I The algebra A = k[x1, x2,..., xn] and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings The module of covariants and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial The setup so far is: ring

Automorphisms I The algebra A = k[x1, x2,..., xn] and endomorphisms I A group G acting on GL(V ) and hence acting as The notions of algebra automorphisms of k[x1, x2,..., xn] invariant subring G Some questions I R = A is the subring comprising invariant polynomials about the invariant subring

Since A is a ring containing R, A is a R-algebra, and in More invariant particular, A is also a R-module. A, viewed as a R-module, subrings Further is termed the module of covariants(deﬁned). connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings Two natural questions and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

Using the setup and notation of the previous question: Automorphisms and I When is the module of covariants free? That is, under endomorphisms what conditions is it true that A is a free R-module? The notions of invariant subring I When is the module of covariants ﬁnitely generated? Some questions about the invariant That is, under what conditions is it true that A is a subring ﬁnitely generated R-module? More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary For instance, in the case of the symmetric group, the algebra of invariants is freely generated by the elementary symmetric polynomials, and the module of covariants is free (the latter is not at all obvious). Kostant looked at the problem of freeness of the module of covariants for the module of covariants, in the case of a connected inﬁnite group, and came up with certain suﬃcient conditions.

Polynomial rings Relating covariants with invariants and their automorphisms

Vipul Naik

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Kostant looked at the problem of freeness of the module of covariants for the module of covariants, in the case of a connected inﬁnite group, and came up with certain suﬃcient conditions.

Polynomial rings Relating covariants with invariants and their automorphisms

Vipul Naik

A crash course in ring theory A remarkable result states that for representations of ﬁnite The polynomial groups, the module of covariants is free if and only if the ring Automorphisms algebra of invariants is free (as an algebra). and For instance, in the case of the symmetric group, the algebra endomorphisms The notions of of invariants is freely generated by the elementary symmetric invariant subring

polynomials, and the module of covariants is free (the latter Some questions about the invariant is not at all obvious). subring

More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings Relating covariants with invariants and their automorphisms

Vipul Naik

polynomials, and the module of covariants is free (the latter Some questions about the invariant is not at all obvious). subring

Kostant looked at the problem of freeness of the module of More invariant covariants for the module of covariants, in the case of a subrings Further connected infinite group, and came up with certain sufficient connections The module of conditions. covariants Harmonic polynomials and the Laplacian A summary In fact, this gives an isomorphism between the polynomial ring in n variables and the ring of partial linear differential operators of order upto n, with multiplication being composition (note that this is a commutative ring because partials commute).

Polynomial rings The diﬀerential operator corresponding to a and their automorphisms polynomial Vipul Naik

A crash course in ring theory

The polynomial ring Given any polynomial p in variables x , x ,..., x , we can Automorphisms 1 2 n and associate a corresponding linear diﬀerential operator, endomorphisms ∂ obtained by replacing each xi by the expression . The notions of ∂xi invariant subring Some questions about the invariant subring

More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings The diﬀerential operator corresponding to a and their automorphisms polynomial Vipul Naik

A crash course in ring theory

operators of order upto n, with multiplication being More invariant composition (note that this is a commutative ring because subrings Further partials commute). connections The module of covariants Harmonic polynomials and the Laplacian A summary This invariant subring will correspond, via the isomorphism, to the invariant subring for the polynomial ring.

Polynomial rings Invariant diﬀerential operators and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

Via the mapping between the polynomial ring and the ring Automorphisms and of diﬀerential operators, we can thus obtain an action of endomorphisms

More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings Invariant diﬀerential operators and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring

Via the mapping between the polynomial ring and the ring Automorphisms and of diﬀerential operators, we can thus obtain an action of endomorphisms

GL(V ) on the ring of linear diﬀerential operators of order The notions of upto n. We can thus also talk of the subring of invariant invariant subring Some questions diﬀerential operators under a given G ≤ GL(V ). about the invariant This invariant subring will correspond, via the isomorphism, subring More invariant to the invariant subring for the polynomial ring. subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Correspondingly, the ring of invariant diﬀerential operators under the action of the orthogonal group is generated by the diﬀerential operator:

X ∂2 ∆ = ∂x2 i i This is the famous Laplacian.

Polynomial rings The particular case of the orthogonal group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial The ring of invariant polynomials under the action of the ring P 2 orthogonal group is generated by i xi . Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings The particular case of the orthogonal group and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial The ring of invariant polynomials under the action of the ring P 2 orthogonal group is generated by i xi . Automorphisms and Correspondingly, the ring of invariant diﬀerential operators endomorphisms

under the action of the orthogonal group is generated by the The notions of diﬀerential operator: invariant subring Some questions 2 about the invariant X ∂ subring ∆ = ∂x2 More invariant i i subrings Further This is the famous Laplacian. connections The module of covariants Harmonic polynomials and the Laplacian A summary I More generally, any multilinear polynomial is harmonic. In fact, the partial derivative in each of the xi s for a multilinear polynomial, is zero (note that the property of being multilinear is not invariant under the action of GL(V ), though the property of being linear is)

Polynomial rings Harmonic polynomials and their automorphisms

Vipul Naik A polynomial f in n variables is termed a harmonic polynomial(deﬁned) if its Laplacian is zero. That is, f is A crash course in ring theory

harmonic if the polynomial: The polynomial ring n ∂2f Automorphisms X and ∂x2 endomorphisms i=1 i The notions of is identically the zero polynomial. invariant subring Some questions Some examples of harmonic polynomials: about the invariant subring I Any linear polynomial is harmonic. More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings Harmonic polynomials and their automorphisms

Vipul Naik A polynomial f in n variables is termed a harmonic polynomial(deﬁned) if its Laplacian is zero. That is, f is A crash course in ring theory

I More generally, any multilinear polynomial is harmonic. Further In fact, the partial derivative in each of the xi s for a connections The module of covariants multilinear polynomial, is zero (note that the property Harmonic polynomials of being multilinear is not invariant under the action of and the Laplacian A summary GL(V ), though the property of being linear is) ∂ Now, the mapping xi 7→ gives an isomorphism: ∂xi

D : A =∼ A˜ Under this identiﬁcation, we get in essence a map:

D1 : A × A → A which is k-bilinear.

Polynomial rings A bilinear map and their automorphisms

Vipul Naik Let A denote the ring of polynomials, and A˜ denote the ring A crash course in of diﬀerential operators. Since any diﬀerential operator acts ring theory

on a polynomial and outputs a polynomial, we have a map: The polynomial ring

Automorphisms A˜ × A → A and endomorphisms

This map is a k-bilinear map, that is, it is k-linear in both The notions of variables. invariant subring Some questions about the invariant subring

More invariant subrings

Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings A bilinear map and their automorphisms

Vipul Naik Let A denote the ring of polynomials, and A˜ denote the ring A crash course in of diﬀerential operators. Since any diﬀerential operator acts ring theory

on a polynomial and outputs a polynomial, we have a map: The polynomial ring

Automorphisms A˜ × A → A and endomorphisms

This map is a k-bilinear map, that is, it is k-linear in both The notions of variables. invariant subring ∂ Some questions Now, the mapping xi 7→ gives an isomorphism: about the invariant ∂xi subring

More invariant D : A =∼ A˜ subrings Further Under this identiﬁcation, we get in essence a map: connections The module of covariants Harmonic polynomials and the Laplacian D1 : A × A → A A summary which is k-bilinear. Polynomial rings Harmonic space as the orthogonal complement and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial Given a subring R of the polynomial ring A, we deﬁne the ring Automorphisms associated harmonic space H as follows: it is the set of and polynomials in A that are annihilated by R under the map endomorphisms The notions of D1. invariant subring This is a k-vector space by the bilinearity of the map. Some questions about the invariant The harmonic polynomials that we saw earlier were the subring elements in the harmonic space corresponding to the More invariant Laplacian. subrings Further connections The module of covariants Harmonic polynomials and the Laplacian A summary Polynomial rings Outline and their automorphisms

Further connections The module of covariants Harmonic polynomials and the Laplacian

A summary I A group G acting on A G I The invariant subring R = A of A under the action of G We considered these questions:

I As a k-algebra, is R free and is it ﬁnitely generated?

I As a R-module, is A free and is it ﬁnitely generated?

Polynomial rings The overall setup and their automorphisms

Vipul Naik

A crash course in ring theory

We were looking at: The polynomial ring

I The ring A of polynomials over k in n variables Automorphisms and endomorphisms

The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary G I The invariant subring R = A of A under the action of G We considered these questions:

I As a k-algebra, is R free and is it ﬁnitely generated?

I As a R-module, is A free and is it ﬁnitely generated?

Polynomial rings The overall setup and their automorphisms

Vipul Naik

A crash course in ring theory

We were looking at: The polynomial ring

I The ring A of polynomials over k in n variables Automorphisms and I A group G acting on A endomorphisms The notions of invariant subring

Some questions about the invariant subring

More invariant subrings

Further connections

A summary I As a R-module, is A free and is it ﬁnitely generated?

Polynomial rings The overall setup and their automorphisms

Vipul Naik

A crash course in ring theory

We were looking at: The polynomial ring

I The ring A of polynomials over k in n variables Automorphisms and I A group G acting on A endomorphisms G The notions of I The invariant subring R = A of A under the action of invariant subring

G Some questions about the invariant We considered these questions: subring More invariant subrings I As a k-algebra, is R free and is it ﬁnitely generated? Further connections

A summary Polynomial rings The overall setup and their automorphisms

Vipul Naik

A crash course in ring theory

We were looking at: The polynomial ring

I The ring A of polynomials over k in n variables Automorphisms and I A group G acting on A endomorphisms G The notions of I The invariant subring R = A of A under the action of invariant subring

G Some questions about the invariant We considered these questions: subring More invariant subrings I As a k-algebra, is R free and is it finitely generated? Further I As a R-module, is A free and is it finitely generated? connections A summary We also used the fact that every polynomial can be naturally identified with a corresponding differential operator, and used this to construct a bilinear map from the space of polynomials to itself.

Polynomial rings The tools we used and their automorphisms

Vipul Naik

A crash course in ring theory

More invariant subrings

Further connections

A summary Polynomial rings The tools we used and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial While studying this question, one useful approach was to ring n think of G acting on k with a certain orbit decomposition, Automorphisms n and and to view the polynomials as functions of k . In particular, endomorphisms this forced the invariant polynomials to become constant The notions of functions on each orbit. invariant subring Some questions We also used the fact that every polynomial can be naturally about the invariant subring identiﬁed with a corresponding diﬀerential operator, and More invariant used this to construct a bilinear map from the space of subrings polynomials to itself. Further connections

A summary I For the orthogonal group, we saw that the invariant subring is a free algebra with generating set being the sum of squares polynomial

Polynomial rings The particular cases and their automorphisms

Vipul Naik

A crash course in ring theory

Some questions about the invariant subring

More invariant subrings

Further connections

A summary Polynomial rings The particular cases and their automorphisms

Vipul Naik

A crash course in ring theory

The polynomial ring For the symmetric group, we saw that the invariant Automorphisms I and subring is a free algebra with generating set being the endomorphisms elementary symmetric polynomials. The notions of invariant subring I For the orthogonal group, we saw that the invariant Some questions about the invariant subring is a free algebra with generating set being the subring sum of squares polynomial More invariant subrings

Further connections

A summary Polynomial rings The particular cases and their automorphisms

Vipul Naik

A crash course in ring theory

Further connections

A summary