Polynomial rings and their automorphisms - Chennai Mathematical

Polynomial ringsand their

automorphisms

Vipul Naik

A crash course inring theory

The polynomialring

Automorphismsandendomorphisms

The notions ofinvariant subring

Some questionsabout the invariantsubring

More invariantsubrings

Furtherconnections

A summary

Polynomial rings and their automorphisms

Vipul Naik

April 23, 2007


automorphisms

Vipul Naik


Definition of ring

Modules over rings

Generating sets andbases

Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Outline

A crash course in ring theoryDefinition of ringModules over ringsGenerating sets and basesRings and idealsConcept of subring

The polynomial ringThe polynomial ring in one variableThe polynomial ring in many variables

Automorphisms and endomorphismsHomomorphism of ringsHomomorphisms from the polynomial ringLinear and affine endomorphisms

The notions of invariant subringThe fixed-point relationship

Some questions about the invariant subringRepresentations and faithful representationsGenerating sets and questions

More invariant subringsThe orthogonal groupRelation between invariant polynomials and vanishing sets

Further connectionsThe module of covariantsHarmonic polynomials and the Laplacian

A summary


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

What’s a ring

A ring is a set R equipped with two binary operations +(addition) and ∗ (multiplication) such that:

I (R,+) forms an Abelian group

I (R, ∗) forms a semigroup (that is, ∗ is an associativebinary 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 theinverse operation is denoted by the prefix unary −.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Ring with identity

A ring with identity is a ring for which the multiplicationoperation has an identity element, that is, there exists anelement 1 ∈ R such that:

a ∗ 1 = 1 ∗ a = a ∀ a ∈ R

In other words, the multiplication operation is a monoidoperation.Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with1 = 0 must be the trivial ring.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Ring with identity


a ∗ 1 = 1 ∗ a = a ∀ a ∈ R

In other words, the multiplication operation is a monoidoperation.

Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with1 = 0 must be the trivial ring.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Ring with identity


a ∗ 1 = 1 ∗ a = a ∀ a ∈ R

In other words, the multiplication operation is a monoidoperation.Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with1 = 0 must be the trivial ring.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Commutative ring

A ring is said to be commutative if the multiplicativeoperation ∗ is commutative.

All the rings we shall be looking at today are so-calledcommutative rings with identity, viz ∗ is commutative andalso has an identity element.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Commutative ring

A ring is said to be commutative if the multiplicativeoperation ∗ is commutative.All the rings we shall be looking at today are so-calledcommutative rings with identity, viz ∗ is commutative andalso has an identity element.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Conventions followed while writing expressions ina ringWe generally adopt the following conventions:

I The multiplication symbol, as well as parentheses formultiplication, are usually omitted. Thus, a ∗ (b ∗ c)may be simply written as abc

I We assume multiplication takes higher precedence overaddition. This helps us leave out a number ofparentheses. For instance, (a ∗ b) + (c ∗ d) can bewritten simply as ab + cd

I Parentheses are also dropped from repeated additionI We denote by n ∈ N the number 1 + 1 + 1 . . . 1 where

we add 1 to itself n times. Moreover, we denote by nxthe number x + x + . . . + x (even when there doesn’texist any 1)

The first of these conventions is justified by associativity ofmultiplication, the second one is justified by the distributivitylaw.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary




I Parentheses are also dropped from repeated addition

I We denote by n ∈ N the number 1 + 1 + 1 . . . 1 wherewe add 1 to itself n times. Moreover, we denote by nxthe number x + x + . . . + x (even when there doesn’texist any 1)



automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary




I Parentheses are also dropped from repeated additionI We denote by n ∈ N the number 1 + 1 + 1 . . . 1 where

we add 1 to itself n times. Moreover, we denote by nxthe number x + x + . . . + x (even when there doesn’texist any 1)



automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Field

A field is a very special kind of ring where the nonzeroelements form a group under multiplication.

Some examples of fields we have seen are Fp (the finite fieldon p elements), Q (the rationals), R (the reals) and C (thecomplex numbers).An example of a ring which is not a field is Z (the ring ofintegers). Another is Z/nZ (the ring of integers modulo n).


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Field

A field is a very special kind of ring where the nonzeroelements form a group under multiplication.Some examples of fields we have seen are Fp (the finite fieldon p elements), Q (the rationals), R (the reals) and C (thecomplex numbers).

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


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Field

A field is a very special kind of ring where the nonzeroelements form a group under multiplication.Some examples of fields we have seen are Fp (the finite fieldon p elements), Q (the rationals), R (the reals) and C (thecomplex numbers).An example of a ring which is not a field is Z (the ring ofintegers). Another is Z/nZ (the ring of integers modulo n).


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Vector space over a field

Let k be a field. A vector space over k is a set V equippedwith a binary operation + and an operation . : k × V → Vsuch that:

I (V ,+) is an Abelian group.

I The map . defines a monoid action of the multiplicativemonoid of k, over V (as Abelian groupautomorphisms). 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 bya 7→ a.v is a group homomorphism


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Module over a ring

Let R be a commutative ring with identity. A module over Ris a set M equipped with a binary operation and a map. : R ×M → M such that:

I (M,+) is an Abelian group.

I The map . defines a monoid action of the multiplicativemonoid of R on M


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Generating set for a module

A set of elements m1,m2, . . . ,mn is said to be a generatingset for a R-module M if given any m ∈ M, we can express mas

∑i rimi where ri ∈ R. The elements mi are termed

generators.

In other words, a generating set is a subset such that everyelement is a R-linear combination of elements from thesubset.A R-module that has a finite generating set is termed afinitely generated R-module.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary




generators.In other words, a generating set is a subset such that everyelement is a R-linear combination of elements from thesubset.

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


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary




generators.In other words, a generating set is a subset such that everyelement is a R-linear combination of elements from thesubset.A R-module that has a finite generating set is termed afinitely generated R-module.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Free generating set

A generating set m1,m2, . . . ,mn for a R-module M istermed a free generating set if:∑

i

rimi = 0 =⇒ ri = 0 ∀ i

In other words, there are no unexpected dependenciesbetween the generators.In general, dependencies of the form

∑i rimi = 0 are termed

relations, and a relation is trivial if and only if all the ri s arezero.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Free generating set


i

rimi = 0 =⇒ ri = 0 ∀ i

In other words, there are no unexpected dependenciesbetween the generators.

In general, dependencies of the form∑

i rimi = 0 are termedrelations, and a relation is trivial if and only if all the ri s arezero.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Free generating set


i

rimi = 0 =⇒ ri = 0 ∀ i

In other words, there are no unexpected dependenciesbetween the generators.In general, dependencies of the form

∑i rimi = 0 are termed

relations, and a relation is trivial if and only if all the ri s arezero.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Irredundant generating set

A generating set m1,m2, . . . ,mn for a R-module M istermed irredundant(defined) if no proper subset of it is agenerating set.

Clearly any free generating set is irredundant, because theability to express one generator as a linear combination ofthe others definitely gives a relation.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Irredundant generating set

A generating set m1,m2, . . . ,mn for a R-module M istermed irredundant(defined) if no proper subset of it is agenerating set.Clearly any free generating set is irredundant, because theability to express one generator as a linear combination ofthe others definitely gives a relation.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

In the case of fields

In the case of fields, the converse is also true. That is, anyirredundant generating set is free. In other words, given anynontrivial relation between the generators, we can expressone of the generators in terms of the others.

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


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

In the case of fields

In the case of fields, the converse is also true. That is, anyirredundant generating set is free. In other words, given anynontrivial relation between the generators, we can expressone of the generators in terms of the others.The idea is to pick any generator with a nonzero coefficient,say ri , and multiply the whole equation by 1/ri , and thentransfer all the other terms to the right side. We can do thisprecisely because every nonzero element is invertible.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Vector spaces and free modules

For a field, every module (viz vector space) has anirredundant generating set, which is also a free generatingset, and in the particular case of fields, we use the termbasis(defined) for such a set.

A module over a ring which possesses a free generating set istermed a free module(defined).


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Vector spaces and free modules

For a field, every module (viz vector space) has anirredundant generating set, which is also a free generatingset, and in the particular case of fields, we use the termbasis(defined) for such a set.A module over a ring which possesses a free generating set istermed a free module(defined).


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Ring as a module over itself

Every ring is a module over itself. In fact, when we’redealing with a ring with identity, it is a free module overitself with the generator being the element 1.

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


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary


Every ring is a module over itself. In fact, when we’redealing with a ring with identity, it is a free module overitself with the generator being the element 1.A submodule of a module is an additive subgroup that isclosed under the ring action.

An ideal(defined) of a ring is a submodule of the ring whenviewed naturally as a module over itself.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary


Every ring is a module over itself. In fact, when we’redealing with a ring with identity, it is a free module overitself with the generator being the element 1.A submodule of a module is an additive subgroup that isclosed under the ring action.An ideal(defined) of a ring is a submodule of the ring whenviewed naturally as a module over itself.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Definition of subring

A subset of a ring is said to be a subring if it is a ring withthe inherited addition and multiplication operations.

In the particular case when the whole ring contains anidentity element, we typically make the following addedassumption about the subring: it contains the identityelement of the whole ring.


automorphisms

Vipul Naik


Definition of ring

Modules over rings


Rings and ideals

Concept of subring

The polynomialring





Furtherconnections

A summary

Definition of subring

A subset of a ring is said to be a subring if it is a ring withthe inherited addition and multiplication operations.In the particular case when the whole ring contains anidentity element, we typically make the following addedassumption about the subring: it contains the identityelement of the whole ring.


automorphisms

Vipul Naik


The polynomialring

The polynomial ringin one variable

The polynomial ringin many variables





Furtherconnections

A summary

Outline








A summary


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Definition of polynomial ring in one variable

Let R be a ring. The polynomial ring in one variable, or oneindeterminate, is the set of all formal polynomials in onevariable, with addition and multiplication defined as usual.

We are in particular interested in the polynomial ring in onevariable over a field.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Definition of polynomial ring in one variable

Let R be a ring. The polynomial ring in one variable, or oneindeterminate, is the set of all formal polynomials in onevariable, with addition and multiplication defined as usual.We are in particular interested in the polynomial ring in onevariable over a field.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Basic properties of the polynomial ring in onevariable

Here are some nice things about the polynomial ring in onevariable over a field:

I If the product of two polynomials is the zeropolynomial, then one of the polynomials must be thezero polynomial. In other words, the product of twononzero polynomials must be a nonzero polynomial.

I The only invertible polynomials in one variable are theconstant nonzero polynomials, viz the nonzeropolynomials of degree zero


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Ideals of the polynomial ring in one variable

If k is a field, then any ideal in k[x ] (viz, anyk[x ]-submodule of k[x ]) is a free module with 1 generator.In other words, it is what is called a principal ideal(defined).

From this, we can in fact deduce the fact that everypolynomial over k[x ] can be written uniquely as a product ofirreducible polynomials, where an irreducible polynomial isone that cannot be factorized further.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Ideals of the polynomial ring in one variable

If k is a field, then any ideal in k[x ] (viz, anyk[x ]-submodule of k[x ]) is a free module with 1 generator.In other words, it is what is called a principal ideal(defined).From this, we can in fact deduce the fact that everypolynomial over k[x ] can be written uniquely as a product ofirreducible polynomials, where an irreducible polynomial isone that cannot be factorized further.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Iterating the polynomial ring operation

For any ring R, we can consider the associated polynomialring R[x1]. Setting this as our new ring, we can consider thenext associated polynomial, R[x1][x2], which is basicallypolynomials with x2 as the indeterminate, over the ringR[x1]. We can do this repeatedly and get something called:

R[x1][x2] . . . [xn]

Now, because of the essentially commutative nature ofthings, we can think of this as simply:

R[x1, x2, . . . , xn]

viz the polynomial ring in n variables


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Nice properties that continue to hold for manyvariables

Let k be a field. Then the n-variate polynomial ring over k,viz k[x1, x2, . . . , xn], satisfies the following:

I The product of any two nonzero polynomials is nonzero

I The only invertible polynomials are the constantnonzero polynomials

I Every polynomial can be factorized uniquely as aproduct of irreducible polynomials (upto factors ofmultiplicative constants)

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


automorphisms

Vipul Naik


The polynomialring


Homomorphism ofrings

Homomorphisms fromthe polynomial ring

Linear and affineendomorphisms




Furtherconnections

A summary

Outline








A summary


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Homomorphism of rings with identity

Let R and S be rings with identity. A homomorphism fromR to S is a map f : R → S such that:

f (a + b) = f (a) + f (b)

f (ab) = f (a)f (b)

f (1) = 1


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Isomorphism, automorphism and endomorphism

We define:

I An isomorphism(defined) is a bijective homomorphism ofrings

I An endomorphism(defined) is a homomorphism from aring to itself (need not be injective, surjective orbijective)

I An automorphism(defined) is an isomorphism from a ringto itself, or equivalently, a bijective endomorphism ofthe ring


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Suffices to locate images of indeterminates

Let k[x1, x2, . . . , xn] be a polynomial ring, and R be anotherring containing a copy of k. Then, the injectivehomomorphism from k to R can be extended to ahomomorphism from k[x1, x2, . . . , xn] to R in many ways.

In fact, for any choice of elements a1, a2, . . . , an ∈ R, thereis a unique homomorphism from k[x1, x2, . . . , xn] to R whichsends each xi to the corresponding ai .


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Suffices to locate images of indeterminates

Let k[x1, x2, . . . , xn] be a polynomial ring, and R be anotherring containing a copy of k. Then, the injectivehomomorphism from k to R can be extended to ahomomorphism from k[x1, x2, . . . , xn] to R in many ways.In fact, for any choice of elements a1, a2, . . . , an ∈ R, thereis a unique homomorphism from k[x1, x2, . . . , xn] to R whichsends each xi to the corresponding ai .


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Endomorphisms of the polynomial ring

Let’s now consider the problem of describing allhomomorphisms from the polynomial ring k[x1, x2, . . . , xn] toitself, which restrict to the identity on k.

Here, the ring R is k[x1, x2, . . . , xn] itself. Hence, to specifythe endomorphism, we need to give polynomialsp1, p2, . . . , pn (each being a polynomial in all the xi s) suchthat each xi maps to the corresponding pi .Thus, every endomorphism of the polynomial ring (that fixesthe base field pointwise) can be described by an arbitrarysequence of n polynomials in the indeterminates.


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary


Let’s now consider the problem of describing allhomomorphisms from the polynomial ring k[x1, x2, . . . , xn] toitself, which restrict to the identity on k.Here, the ring R is k[x1, x2, . . . , xn] itself. Hence, to specifythe endomorphism, we need to give polynomialsp1, p2, . . . , pn (each being a polynomial in all the xi s) suchthat each xi maps to the corresponding pi .

Thus, every endomorphism of the polynomial ring (that fixesthe base field pointwise) can be described by an arbitrarysequence of n polynomials in the indeterminates.


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary


Let’s now consider the problem of describing allhomomorphisms from the polynomial ring k[x1, x2, . . . , xn] toitself, which restrict to the identity on k.Here, the ring R is k[x1, x2, . . . , xn] itself. Hence, to specifythe endomorphism, we need to give polynomialsp1, p2, . . . , pn (each being a polynomial in all the xi s) suchthat each xi maps to the corresponding pi .Thus, every endomorphism of the polynomial ring (that fixesthe base field pointwise) can be described by an arbitrarysequence of n polynomials in the indeterminates.


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Composing two endomorphisms

The rule for composing endomorphisms of the polynomialring is as follows: If the endomorphisms arep = (p1, p2, . . . , pn) and q = (q1, q2, . . . , qn) then theircomposite q ◦ p is the endomorphism

xi 7→ qi (p1(x1, x2, . . . , xn), p2(x1, x2, . . . , xn), . . . , pn(x1, x2, . . . , xn))

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


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Endomorphisms of the polynomial ring in onevariable

In the case of k[x ] (polynomial ring in one variable), theendomorphisms are described simply by polynomials.Composition of endomorphisms is, in this context,composition of polynomials.

It’s clear that the only polynomials which have an inverse inthe composition sense are the linear polynomials. In otherwords, the automorphism group of the polynomial ring inone variable is the group of affine maps x 7→ ax + b.


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Endomorphisms of the polynomial ring in onevariable

In the case of k[x ] (polynomial ring in one variable), theendomorphisms are described simply by polynomials.Composition of endomorphisms is, in this context,composition of polynomials.It’s clear that the only polynomials which have an inverse inthe composition sense are the linear polynomials. In otherwords, the automorphism group of the polynomial ring inone variable is the group of affine maps x 7→ ax + b.


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

Affine endomorphisms in more than one variable

Given k[x1, x2, . . . , xn] we can consider endomorphismswhere all the pi s are linear polynomials in the xi s. Thiscorresponds to affine maps on the vector space kn (basisvectors viewed as xi s)

More specifically, if we consider endomorphisms where all thepi s are homogeneous linear polynomials (viz, linearpolynomials without a constant term), we get somethingwhich corresponds to linear maps on the vector space kn

(basis vectors viewed as xi s)Among these, the invertible elements are precisely thosewhich correspond to invertible affine (respectively linear)maps – viz GAn(k) (respectively GLn(k)).


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary


Given k[x1, x2, . . . , xn] we can consider endomorphismswhere all the pi s are linear polynomials in the xi s. Thiscorresponds to affine maps on the vector space kn (basisvectors viewed as xi s)More specifically, if we consider endomorphisms where all thepi s are homogeneous linear polynomials (viz, linearpolynomials without a constant term), we get somethingwhich corresponds to linear maps on the vector space kn

(basis vectors viewed as xi s)

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


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary


Given k[x1, x2, . . . , xn] we can consider endomorphismswhere all the pi s are linear polynomials in the xi s. Thiscorresponds to affine maps on the vector space kn (basisvectors viewed as xi s)More specifically, if we consider endomorphisms where all thepi s are homogeneous linear polynomials (viz, linearpolynomials without a constant term), we get somethingwhich corresponds to linear maps on the vector space kn

(basis vectors viewed as xi s)Among these, the invertible elements are precisely thosewhich correspond to invertible affine (respectively linear)maps – viz GAn(k) (respectively GLn(k)).


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

The upshot

The upshot is that:

GLn(k) ≤ GAn(k) ≤ Aut(k[x1, x2, . . . , xn])

In other words, every linear automorphism (more generallyevery affine automorphism) gives rise to a polynomialautomorphism, and this association is faithful.

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


automorphisms

Vipul Naik


The polynomialring








Furtherconnections

A summary

The upshot

The upshot is that:

GLn(k) ≤ GAn(k) ≤ Aut(k[x1, x2, . . . , xn])

In other words, every linear automorphism (more generallyevery affine automorphism) gives rise to a polynomialautomorphism, and this association is faithful.There’s something nice about the polynomial automorphismsthat come from linear automorphisms. Namely, theseautomorphisms actually preserve the degree of thepolynomial. Any automorphism that is not linear will notpreserve the degree of at least some polynomial.


automorphisms

Vipul Naik


The polynomialring



The fixed-pointrelationship



Furtherconnections

A summary

Outline








A summary


automorphisms

Vipul Naik


The polynomialring






Furtherconnections

A summary

The fixed-point relationship

Given an automorphism σ of the polynomial ring, and apolynomial p that sits inside this polynomial ring, we saythat p is a fixed point of σ if σ(p) = p.

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


automorphisms

Vipul Naik


The polynomialring






Furtherconnections

A summary


Given an automorphism σ of the polynomial ring, and apolynomial p that sits inside this polynomial ring, we saythat p is a fixed point of σ if σ(p) = p.Thus, given any set P of polynomials p, we can consider theset of all automorphisms σ for which every p ∈ P is a fixedpoint. This set of automorphisms is clearly a subgroup of theautomorphism group.

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


automorphisms

Vipul Naik


The polynomialring






Furtherconnections

A summary


Given an automorphism σ of the polynomial ring, and apolynomial p that sits inside this polynomial ring, we saythat p is a fixed point of σ if σ(p) = p.Thus, given any set P of polynomials p, we can consider theset of all automorphisms σ for which every p ∈ P is a fixedpoint. This set of automorphisms is clearly a subgroup of theautomorphism group.Analogously, for every subset S of the automorphism group,we can consider the set Q of all polynomials that are fixedpoints of every σ ∈ S . Note that this set Q is clearly a ring.


automorphisms

Vipul Naik


The polynomialring






Furtherconnections

A summary

Notion of Galois correspondence

The above can be fitted into the framework of a Galoiscorrespondence.Given two sets A and B and a relation R between A and B,the Galois correspondence for R is a pair of mapsS : 2A → 2B and T : 2B → 2A defined as:

I For C ≤ A, S(C ) is the set of all elements in B that arerelated to every element in C

I For D ≤ B, T (D) is the set of all elements in A thatare related to every element in D

Then we have:

I C1 ⊆ C2 =⇒ S(C2) ⊆ S(C1) and similarly for T

I S ◦ T ◦ S = S and T ◦ S ◦ T = T


automorphisms

Vipul Naik


The polynomialring






Furtherconnections

A summary

How this fits in

In our case, the relation is the fixed-point relation. That is,the two sets are:

I A is the set of all polynomials

I B is the group GL(V )

I R is the relation of the given polynomial

We are interested in taking subgroups of GL(V ) and askingfor the invariant subrings, or conversely, in taking subrings ofA and asking for the fixing subgroups. This essentiallycorresponds to computing the maps T and S .


automorphisms

Vipul Naik


The polynomialring




Representations andfaithfulrepresentations

Generating sets andquestions


Furtherconnections

A summary

Outline








A summary


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Representation of a group

Let G be a group. A linear representation(defined) of G overa field k is a homomorphism ρ : G → GL(V ) where V is avector space over k and GL(V ) is the group of k-linearautomorphisms of V .

The representation is said to be faithful(defined) if ρ is aninjective map. In other words, we view G as a subgroup ofGL(V ).


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Representation of a group

Let G be a group. A linear representation(defined) of G overa field k is a homomorphism ρ : G → GL(V ) where V is avector space over k and GL(V ) is the group of k-linearautomorphisms of V .The representation is said to be faithful(defined) if ρ is aninjective map. In other words, we view G as a subgroup ofGL(V ).


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Invariant subring for a representation

One of the many aspects to a representation of a group isthe following: What is the subring of polynomials that areinvariant under the action of the group? In other words,what are the polynomials that are unchanged under theaction of the group on the xi s?

This is in essence the same as the question we consideredearlier.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Invariant subring for a representation

One of the many aspects to a representation of a group isthe following: What is the subring of polynomials that areinvariant under the action of the group? In other words,what are the polynomials that are unchanged under theaction of the group on the xi s?This is in essence the same as the question we consideredearlier.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Symmetric polynomials

A polynomial is said to be a symmetric polynomial(defined) ifit remains unchanged under any permutation of the xi s.Clearly, the symmetric polynomials form a subring of the ringof all polynomials.

This is precisely the same as the ring of invariant polynomialscorresponding to the symmetric group embedded naturallyas permutations of the basis elements, in GLn(k).


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Symmetric polynomials

A polynomial is said to be a symmetric polynomial(defined) ifit remains unchanged under any permutation of the xi s.Clearly, the symmetric polynomials form a subring of the ringof all polynomials.This is precisely the same as the ring of invariant polynomialscorresponding to the symmetric group embedded naturallyas permutations of the basis elements, in GLn(k).


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Vipul Naik


The polynomialring







Furtherconnections

A summary

Elementary symmetric polynomials

The elementary symmetric polynomial of degree j overvariables x1, x2, . . . , xn is defined as the coefficient of xn−j inthe expression: ∏

i

(x + xi )

Or equivalently, as (−1)j times the coefficient of xn−j in theexpression

∏i (x − xi ).

We shall use the letter sj to denote the elementarysymmetric polynomial of degree j .


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Two remarkable facts

It is clear that any elementary symmetric polynomial is asymmetric polynomial. Thus, any polynomial in terms of theelementary symmetric polynomials also is an elementarysymmetric polynomial. In other words, we have ahomomorphism:

k[s1, s2, . . . , sn] → k[x1, x2, . . . , xn]Sn

Two remarkable facts are:

I This mapping is injective. That is, any two differentpolynomials in the sjs give rise to different polynomialsin the xi s.

I This mapping is surjective. That is, any symmetricpolynomial in the xi s can be expressed as a polynomialin the sjs.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Generating set for an algebra

What we have done is shown that the invariant subring forthe symmetric group is in fact itself isomorphic to apolynomial ring, in other words, we can find polynomials init such that this subring is generated by these polynomials,without any further relations between them.

This gives some notions. Let k be a base field. Then anyring R containing k is termed a k-algebra. A generating setfor R is a set S such that every element of R can beexpressed as a polynomial in elements of S with coefficientsfrom k.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Generating set for an algebra

What we have done is shown that the invariant subring forthe symmetric group is in fact itself isomorphic to apolynomial ring, in other words, we can find polynomials init such that this subring is generated by these polynomials,without any further relations between them.This gives some notions. Let k be a base field. Then anyring R containing k is termed a k-algebra. A generating setfor R is a set S such that every element of R can beexpressed as a polynomial in elements of S with coefficientsfrom k.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Generating set (continued)

I An algebra over k is said to be finitely generated(defined)

if it has a finite generating set as a k-algebra, that is,there is a surjective homomorphism to it from thepolynomial ring in finitely many variables

I An algebra over k is said to be free(defined) if we can finda generating set such that the mapping from thepolynomial ring of that generating set to the givenalgebra, is an isomorphism.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Two questions of interest

Given a group G and a (without loss of generality, faithful)linear representation of G of degree n, letR = k[x1, x2, . . . , xn]

G be the invariant subringcorresponding to G . Two questions we are interested in are:

I Is R a finitely generated k-algebra?

I Is R a free k-algebra? That is, can R be viewed as thepolynomial ring in some number of variables?

In the case where G is the symmetric group, the answer toboth questions was yes, the elementary symmetricpolynomials formed a finite freely generating set for R.


automorphisms

Vipul Naik


The polynomialring





The orthogonal group

Relation betweeninvariant polynomialsand vanishing sets

Furtherconnections

A summary

Outline








A summary


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Definition of the orthogonal group

The orthogonal group of order n over a field k, denoted asOn(k), is defined as the group of those matrices A such thatAAT is the identity matrix.

Equivalently, it is the group of those transformation of thespace kn that fix the origin and preserve the norm of anyvector, that is, they preserve

∑i x

2i for any vector

(x1, x2, . . . , xn).Equivalently, it is the group of those transformations of thespace kn that preserve the scalar product of any two vectors.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Definition of the orthogonal group

The orthogonal group of order n over a field k, denoted asOn(k), is defined as the group of those matrices A such thatAAT is the identity matrix.Equivalently, it is the group of those transformation of thespace kn that fix the origin and preserve the norm of anyvector, that is, they preserve

∑i x

2i for any vector

(x1, x2, . . . , xn).Equivalently, it is the group of those transformations of thespace kn that preserve the scalar product of any two vectors.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Invariant polynomials for the orthogonal group

Clearly, the polynomial∑

i x2i is an invariant polynomial

under the action of the orthogonal group. Hence, theinvariant subring contains, as a subring, the polynomial ringgenerated by

∑i x

2i .

It turns out that the converse is also true: any polynomial inthe xi s that is invariant under the action of the orthogonalgroup must actually be a polynomial in

∑i x

2i .


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

A closer inspection of the orthogonal groupaction

The orthogonal group acts on the space kn, and kn naturallydecomposes into orbits under the action. Since everyelement of the orthogonal group preserves the polynomial∑

i x2i , each orbit must lie inside a “sphere” of the form∑

i x2i = c for some value of c .

It turns out that the action is also transitive, i.e. given anytwo points on the same sphere, there is an element of theorthogonal group taking one to the other. This essentiallyfollows from the fact that any unit vector can be completedto an orthonormal basis.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

The proof for the invariant subring

Proving that the invariant subring comprises polynomials in∑i x

2i thus reduces to proving that:

Any polynomial that is constant on spheres (that is, loci ofthe form

∑i x

2i = c) must be a polynomial in

∑i x

2i .

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


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Proof of the general statement

We write x for the tuple (x1, x2, . . . , xn).Consider the locus p(x) = c . Suppose f (x) takes the valueλ on this locus. Then, by the factor theorem:

f (x)− λ = h(x)(p(x)− c)

where h(x) is another polynomial.

Now, h also satisfies the property of being constant on everylocus p(x) = c ′. By induction, we can write h as apolynomial in p, and hence f can also be written as apolynomial in p.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Proof of the general statement

We write x for the tuple (x1, x2, . . . , xn).Consider the locus p(x) = c . Suppose f (x) takes the valueλ on this locus. Then, by the factor theorem:

f (x)− λ = h(x)(p(x)− c)

where h(x) is another polynomial.Now, h also satisfies the property of being constant on everylocus p(x) = c ′. By induction, we can write h as apolynomial in p, and hence f can also be written as apolynomial in p.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Upshot: for the orthogonal group

We have shown that for the orthogonal group, the invariantsubring is in fact the polynomial ring in one variable. Hence,it is both free and finitely generated.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Orbits as sets of constancy

If G ≤ GL(V ), then any orbit of kn under the action of G ,must take a constant value under any polynomial invariantunder the action of G . In other words, we can define tworelations:

I Given a subring R of the polynomial ring, call x , y ∈ kn

as R-equivalent if f (x) = f (y) for any f ∈ R

I Given a group G ≤ GL(V ), call x , y ∈ kn asG -equivalent if there exists g ∈ G such that g .x = y

Then if R is the invariant subring for G , G -equivalenceimplies R-equivalence.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Rings of constant functions versus ideals

Given a subset S ⊆ kn, define I (S) as the set of allpolynomials that vanish at every point of S , and R(S) as thering of all polynomials that are constant on S . Then:

R(S) = I (S) + k

In other words, every polynomial constant on S can bewritten as a polynomial that vanishes on S , plus a constantpolynomials.


automorphisms

Vipul Naik


The polynomialring







Furtherconnections

A summary

Expression for the invariant subring

Here’s the chain of reasoning:

I Any polynomial invariant under the action of G must beconstant on all the G -orbits

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

k[x1, x2, . . . , xn]G =

⋂O

R(O)

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

k[x1, x2, . . . , xn]G =

⋂O

I (O) + k


automorphisms

Vipul Naik


The polynomialring





Furtherconnections

The module ofcovariants

Harmonic polynomialsand the Laplacian

A summary

Outline








A summary


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

The module of covariants

The setup so far is:

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

I A group G acting on GL(V ) and hence acting asalgebra automorphisms of k[x1, x2, . . . , xn]

I R = AG is the subring comprising invariant polynomials

Since A is a ring containing R, A is a R-algebra, and inparticular, A is also a R-module. A, viewed as a R-module,is termed the module of covariants(defined).


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

Two natural questions

Using the setup and notation of the previous question:

I When is the module of covariants free? That is, underwhat conditions is it true that A is a free R-module?

I When is the module of covariants finitely generated?That is, under what conditions is it true that A is afinitely generated R-module?


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

Relating covariants with invariants

A remarkable result states that for representations of finitegroups, the module of covariants is free if and only if thealgebra of invariants is free (as an algebra).

For instance, in the case of the symmetric group, the algebraof invariants is freely generated by the elementary symmetricpolynomials, and the module of covariants is free (the latteris not at all obvious).Kostant looked at the problem of freeness of the module ofcovariants for the module of covariants, in the case of aconnected infinite group, and came up with certain sufficientconditions.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary


A remarkable result states that for representations of finitegroups, the module of covariants is free if and only if thealgebra of invariants is free (as an algebra).For instance, in the case of the symmetric group, the algebraof invariants is freely generated by the elementary symmetricpolynomials, and the module of covariants is free (the latteris not at all obvious).

Kostant looked at the problem of freeness of the module ofcovariants for the module of covariants, in the case of aconnected infinite group, and came up with certain sufficientconditions.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary


A remarkable result states that for representations of finitegroups, the module of covariants is free if and only if thealgebra of invariants is free (as an algebra).For instance, in the case of the symmetric group, the algebraof invariants is freely generated by the elementary symmetricpolynomials, and the module of covariants is free (the latteris not at all obvious).Kostant looked at the problem of freeness of the module ofcovariants for the module of covariants, in the case of aconnected infinite group, and came up with certain sufficientconditions.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

The differential operator corresponding to apolynomial

Given any polynomial p in variables x1, x2, . . . , xn, we canassociate a corresponding linear differential operator,obtained by replacing each xi by the expression ∂

∂xi.

In fact, this gives an isomorphism between the polynomialring in n variables and the ring of partial linear differentialoperators of order upto n, with multiplication beingcomposition (note that this is a commutative ring becausepartials commute).


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

Invariant differential operators

Via the mapping between the polynomial ring and the ringof differential operators, we can thus obtain an action ofGL(V ) on the ring of linear differential operators of orderupto n. We can thus also talk of the subring of invariantdifferential operators under a given G ≤ GL(V ).

This invariant subring will correspond, via the isomorphism,to the invariant subring for the polynomial ring.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

Invariant differential operators

Via the mapping between the polynomial ring and the ringof differential operators, we can thus obtain an action ofGL(V ) on the ring of linear differential operators of orderupto n. We can thus also talk of the subring of invariantdifferential operators under a given G ≤ GL(V ).This invariant subring will correspond, via the isomorphism,to the invariant subring for the polynomial ring.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

The particular case of the orthogonal group

The ring of invariant polynomials under the action of theorthogonal group is generated by

∑i x

2i .

Correspondingly, the ring of invariant differential operatorsunder the action of the orthogonal group is generated by thedifferential operator:

∆ =∑

i

∂2

∂x2i

This is the famous Laplacian.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

Harmonic polynomials

A polynomial f in n variables is termed a harmonicpolynomial(defined) if its Laplacian is zero. That is, f isharmonic if the polynomial:

n∑i=1

∂2f

∂x2i

is identically the zero polynomial.Some examples of harmonic polynomials:

I Any linear polynomial is harmonic.

I More generally, any multilinear polynomial is harmonic.In fact, the partial derivative in each of the xi s for amultilinear polynomial, is zero (note that the propertyof being multilinear is not invariant under the action ofGL(V ), though the property of being linear is)


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

A bilinear map

Let A denote the ring of polynomials, and A denote the ringof differential operators. Since any differential operator actson a polynomial and outputs a polynomial, we have a map:

A× A → A

This map is a k-bilinear map, that is, it is k-linear in bothvariables.

Now, the mapping xi 7→ ∂∂xi

gives an isomorphism:

D : A ∼= A

Under this identification, we get in essence a map:

D1 : A× A → A

which is k-bilinear.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

A bilinear map

Let A denote the ring of polynomials, and A denote the ringof differential operators. Since any differential operator actson a polynomial and outputs a polynomial, we have a map:

A× A → A

This map is a k-bilinear map, that is, it is k-linear in bothvariables.Now, the mapping xi 7→ ∂

∂xigives an isomorphism:

D : A ∼= A

Under this identification, we get in essence a map:

D1 : A× A → A

which is k-bilinear.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections



A summary

Harmonic space as the orthogonal complement

Given a subring R of the polynomial ring A, we define theassociated harmonic space H as follows: it is the set ofpolynomials in A that are annihilated by R under the mapD1.This is a k-vector space by the bilinearity of the map.The harmonic polynomials that we saw earlier were theelements in the harmonic space corresponding to theLaplacian.


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Vipul Naik


The polynomialring





Furtherconnections

A summary

Outline








A summary


automorphisms

Vipul Naik


The polynomialring





Furtherconnections

A summary

The overall setup

We were looking at:

I The ring A of polynomials over k in n variables

I A group G acting on A

I The invariant subring R = AG of A under the action ofG

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?


automorphisms

Vipul Naik


The polynomialring





Furtherconnections

A summary

The tools we used

While studying this question, one useful approach was tothink of G acting on kn with a certain orbit decomposition,and to view the polynomials as functions of kn. In particular,this forced the invariant polynomials to become constantfunctions on each orbit.

We also used the fact that every polynomial can be naturallyidentified with a corresponding differential operator, andused this to construct a bilinear map from the space ofpolynomials to itself.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections

A summary

The tools we used

While studying this question, one useful approach was tothink of G acting on kn with a certain orbit decomposition,and to view the polynomials as functions of kn. In particular,this forced the invariant polynomials to become constantfunctions on each orbit.We also used the fact that every polynomial can be naturallyidentified with a corresponding differential operator, andused this to construct a bilinear map from the space ofpolynomials to itself.


automorphisms

Vipul Naik


The polynomialring





Furtherconnections

A summary

The particular cases

I For the symmetric group, we saw that the invariantsubring is a free algebra with generating set being theelementary symmetric polynomials.

I For the orthogonal group, we saw that the invariantsubring is a free algebra with generating set being thesum of squares polynomial

Polynomial rings and their automorphisms - Chennai Mathematical

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Transcript of Polynomial rings and their automorphisms - Chennai Mathematical