Flag Varieties and...

Flag Varieties and Representations

Lectures by Aaron BertramPCMI 2015 Undergraduate Summer School

Notes by David [email protected]

Contents

Lecture 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Lecture 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Lecture 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Lecture 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Lecture 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Lecture 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Lecture 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Lecture 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Lecture 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Lecture 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Lecture 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Lecture 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Lecture 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Lecture 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1

mailto:[email protected]

Lecture 1 June 30, 2015

Yesterday Tom Garrity said that “functions define the world” (and we all learnit as children). The better version is that “categories define our universe”).

Action of a group is always inside a category. We will have a categoricalmoment of zen for each category.

Definition 1. A Category is a collection of objects and a collection of arrows /morphisms. These must satisfy several properties:

1. Morphisms compose correctly. If A fÝÑ B and B

gÝÑ C then A

g˝fÝÝÑ C is

another morphism;

2. Composition is associative;

3. For each objectX there is an identity arrow idX such that for anyX fÝÑ Y ,

f ˝ idX “ f and for any Z gÝÑ X , idX ˝ g “ g.

Definition 2. In any category, A fÝÑ B is an isomorphism if f has an inverse,

i.e. there is a morphism BgÝÑ A such that f ˝ g “ idB and g ˝ f “ idA. An

automorphism is an isomorphism X Ñ X .

Left and right inverses are necessarily equal: if gl and gr are left/ rightinverses, we have

gl “ gl ˝ idB “ gl ˝ f ˝ gr “ idA ˝ gr “ gr.

Definition 3. A Monoid is a category with one element.

Fun fact: a group can be defined as a monoid in which every morphism isan isomorphism. An example of a group is AutpXq “ tautomorphisms of Xufor any object X of any category.

We will talk about several categories, starting with the category of sets, andthen vector spaces, and later the category of groups.

Category of Sets and Permutations

In this category, Sets, the objects are sets and the arrows are functions (but we’llcall them mappings for now). We restrict attention to the category of finite sets,finSets, where the objects are finite sets. There is a sequence of finite sets:

H Ă r1s “ t1u Ă r2s “ t1, 2u Ă . . . Ă rns Ă . . . .

These are all isomorphism classes in finSets.Let’s look at the endomorphisms of these objects in the category finSets.

#Endprnsq “ nn and #Autprnsq “ n!. Note that endomorphisms don’t com-mute, and automorphisms are just permutations. The symmetric group isSn “ Autprnsq. (Which is Permpnq in Aaron’s notes but that’s stupid).

2

Definition 4. Here’s the cycle notation for permutations. For a permutation1 ÞÑ 2, 2 ÞÑ 3, 4 ÞÑ 5, 5 ÞÑ 4, the notation is p1 2 3qp4 5q. Each parentheti-cal bit represents a cycle 1 ÞÑ 2 ÞÑ 3 ÞÑ 1 and 4 ÞÑ 5 ÞÑ 4. The identity isp1qp2qp3qp4qp5q. To make our notation more efficient, we drop singletons.

Example 5.S3 “ tid, p1 2q, p1 3q, p1 2 3q, p1 3 2qu

S4 “

"

id, p1 2q, p1 3q, p1 4q, p2 3q, p2 4q, p3 4q, p1 2 3q, p1 3 2q, p1 2 4q, p1 4 2q,

p1 3 4q, p1 4 3q, p2 3 4q, p2 4 3q, p1 2qp3 4q, p1 3qp2 4q, p1 4qp2 3q

*

Definition 6. The sign of a permutation is defined in fact for any f : rns Ñ rns

by

sgnpfq :“ź

iăj

fpjq ´ fpiq

j ´ i

Proposition 7. Let f : rns Ñ rns.

1. sgnpfq “ 0 unless f is invertible;

2. sgnpσq “ ˘1 if σ P Sn;

3. sgnpσ ˝ τq “ sgnpσq ˝ sgnpτq;

4. sgnpidq “ 1;

5. sgnppi jqq “ ´1.

Example 8. In S3,

sgnpp1 2qq “1´ 2

2´ 1¨

3´ 2

3´ 1¨

3´ 1

3´ 2“ ´1.

sgnpp1 2 3qq “3´ 2

2´ 1¨

1´ 2

3´ 1¨

1´ 3

3´ 2“ 1

Proof of Proposition 7. 1. If f is not a bijection, then it’s not injective, so fpjq “fpiq for some i, j. So there is a term pfpjq´fpiqq{pj´iq “ 0 in the product.Conversely, if f is a bijection then this cannot happen.

2. If f is a bijection, the sign must be ˘1 because the numerator and de-nominator have the same things, but in a different order. This means that|fpiq ´ fpjq| “ |k ´ `| for some k, ` P rns and k ´ ` or `´ k will appear inthe denominator at some point cancelling the fpiq ´ fpjq. This happensfor each i, j.

ś

iăj |fpjq ´ fpiq|ś

iăj |j ´ i|“ 1.

3

3.

sgnpσ ˝ τq “ź

iăj

σpτpiqq ´ σpτpjqq

j ´ i

“ź

iăj

σpτpjqq ´ σpτpiqq

τpjq ´ τpiq

τpjq ´ τpiq

j ´ i

“ź

τpiqăτpjq

σpτpjqq ´ σpτpiqq

τpjq ´ τpiq

ź

iăj

τpjq ´ τpiq

j ´ i“ sgnpσqsgnpτq

4. Clear.

5. Exercise.

Here’s a fun fact. Note that

pa1 a2 a3 . . . anq “ pa1 anq . . . pa1 a3qpa1 a2q,

so we can conclude that any σ P Sn is a product of transpositions.Another fun fact,

ÿ

σP§n

sgnpσq “ 0

Lecture 2 July 1, 2015

I’m going to start with a great line. Almost all algebraic geometry talks beginwith this simple sentence: let k be a field.

Maybe k “ Q, or maybe k “ R, which is the completion of the first option,Q, with respect to the ordinary norm | ¨ |. But we can also complete with respectto the p-adic norm | ¨ |p for p a prime (p is always prime, except on wikipediawhere p “ 10).

Q Ă R

Here’s how the p-adic norm works. For n an integer, |pn|p “ 1pn , |1|p “ 1,

and if ab P Q such that p - a and p - b, then |ab |p “ 0. Completing with respect tothis norm gives the p-adic numbers Qp, also a field. And it too has an algebraicclosure.

Q Ă Qp

Of course, we also have the algebraic closure of R, which is the familiarcomplex numbers C. We can also take the algebraic closure of Q to get thealgebraic numbers Q. Fields between Q and Q are number fields k, if they arefinite extensions of Q.

4

Problem 9. An open and very fundamental question in number theory. Theinverse galois problem: does every finite group appear as the Galois group ofsome field extension k{Q?

There are finite fields too, Fp “ Z{pZ, and Fpn Ą Fp. We can also takealgebraic closures of Fp, to get fields Fp, which are no longer finite.

There are fields even bigger than C as well, like the field of rational func-tions

Cptq “"

pptq

qptq

ˇ

ˇ

ˇ

ˇ

p, q P Crts*

,

or it’s algebraic closure. Or the field of Laurent series

Cpptqq “

#

8ÿ

i“´n

aiti

ˇ

ˇ

ˇ

ˇ

ˇ

ai P C

+

.

It’s algebraic closure is the Puiseaux series Cpptqq. Maybe you have two vari-ables, with a field Cpt1, t2q. Or n variables, such as Cpt1, . . . , tnq. In some senseall of algebraic number theory lives inside the algebraic closure of one of thesethings. And then there are local fields, global fields, perfect fields, etc.

Okay enough about fields. When we say “let k be a field,” we mean thatk “ C or k “ R.

Last time we talked about the category of sets, Sets. Let’s define the productof two sets categorically.

Definition 10. The product of two sets S, T is a set U together with mapsp : U Ñ S and q : U Ñ T such that for any other set Z with maps f : Z Ñ S andg : Z Ñ T , there is a unique map u : Z Ñ U such that p ˝ u “ f and q ˝ u “ g.

Z

U

S T

f gu

p q

Now let’s talk about vector spaces over a fixed field k. We have a categorycalled k-Vect of vector spaces over k, where the objects are k-vector spaces andthe morphisms are linear maps. k--Vect is an abelian category, which in thiscase tells us that

HompV,W q :“ tlinear maps f : V ÑW u

is also a k-vector space. A special case of this is the dual space V ˚ :“ HompV, kq.After we choose a bases, we can think of Hompkm, knq as nˆm matrices.

The hom-sets have more structure still! EndpV q “ HompV, V q is a non-commutative ring of linear transformations (matrices after we choose a basis),

5

and GLpV q “ AutpV q is the invertible elements of HompV, V q, which forms agroup (it’s the group of units of EndpV q).

The standard linear algebra thing to do next is to learn about determinants.The determinant is a map

det : HompV, V q Ñ k,

and it’s well defined because it’s independent of any choice of basis.

Definition 11. We’ll define the determinant for a matrix A “ paijq as

detpAq :“ÿ

σPSn

sgnpσqnź

i“1

aiσpiq.

Next, some representation theory. We’ll commit the cardinal sin of choosinga basis e1, . . . , en for kn.

Definition 12. The permutation representation of Sn on kn corresponds to theaction of Sn on vectors by permuting the coordinates. This is given by a mapSn Ñ GLpknq, σ ÞÑ Pσ , where Pσ is the permutation matrix with columnsgiven by Pσei “ eσpiq.

Problem 13. Given a vector ~v “řni“1 viei P k

n, then what is Pσ~v ?

Note that the determinant of a permutation matrix Pσ is the sign of its per-mutation: detpPσq “ sgnpσq.

Definition 14. A matrix A is semisimple if and only if there is an invertiblematrix L such that LAL´1 is diagonal. This is exactly when A has a basis ofeigenvectors.

Example 15. Let’s find the eigenvalues of a rotation matrix„

cos θ ´ sin θ

sin θ cos θ

.

It has no real eigenvalues. We compute

det

„

λ´ cos θ sin θ

´ sin θ λ´ cos θ

“ pλ´ cos θq2 ` sin2 θ

“ λ2 ´ 2 cos θλ` cos2 θ ` sin2 θ

ùñ λ “2 cos θ ˘

?4´ 4 cos2 θ

2“ cos θ ˘ i sin θ “ eiθ

Let’s find the eigenvectors too. We want the kernel of„

eiθ ´ cos θ sin θ

´ sin θ eiθ ´ cos θ

“

„

i sin θ sin θ

´ sin θ i sin θ

,

6

which is spanned by„

1

í

and„

1

i

,

corresponding to eiθ and eíθ, respectively.


First, a few comments about yesterday’s lecture. At the very end, we said thatin C2, p1, iq K p1,íq. This is not true for the ordinary dot product. We need touse a Hermitian inner product, which caused some confusion.

We also forgot to talk about quotient spaces.

Definition 16. If V is a vector space and U ď V is a subspace, then we canform the quotient vector space

V {U :“ t~v ` U | v P V u.

Moreover, there is a natural, linear, surjective map V�V {U .

This is interesting in representation theory, because one of the problemswe’ll be interested in is lifting a representation of V {U to a representation of V .

Okay, now let’s move onto today’s lecture. We’re going to talk about groupactions. Some examples of groups we’re going to talk about include AutpXq forX an object in some category C, or maybe Autprnsq – Sn, or maybe GLpV q “AutpV q for V a vector space.

Definition 17. The category of groups, which we call Groups, is the cate-gory in which the objects are groups and the morphisms are group homomor-phisms.

Example 18. The exponential map is a group homomorphism from pR,`q topRą0, ¨q, because ex`y “ exey . It has an inverse log, but if we instead had theexponential e : pC,`q Ñ pC˚, ¨q, it only has a local inverse.

Example 19. Another example we saw yesterday, which is the map P : Sn Ñ

GLnk that sends σ P Sn to the permutation matrix Pσ . There is an interestingcommuting diagram of groups.

Sn GLnk

t˘1u k˚

φ

sgn det

Definition 20. A group is finitely presented if it can be described with finitelymany generators and finitely many relations, which we writeB

x1, . . . , xn

ˇ

ˇ

ˇ

ˇ

w1px1, . . . , xnq “ w2px1, . . . , xnq “ . . . “ wmpx1, . . . , xnq “ 1

F

.

7

Example 21. The free group on two generators is given by the finite presenta-tion xx, yy. We can draw it too! It’s a nifty a fractal.

Each finitely presented group is a quotient of a free group by some normalsubgroup that creates the relations.

Example 22. The cyclic group Cn :“ xx | xn “ 1y. The integers, Z – xxy is theinfinite cyclic group.

Example 23. The dihedral group Dn :“ xx, y | xn “ y2 “ xyxy´1 “ 1y. It’s notimmediately clear that this group is finite, but we can list all of the elements as

t1, x, x2, . . . , xn´1, y, xy, . . . , xn´1yu

and then use the presentation to show that these are all of the elements. Hereis another fun fact:

pxyq2 “ xpyxqy “ xpx´1yqy “ 1.

Example 24. The symmetric group is generated by xi :“ pi i` 1q for 1 ď i ď n

according to the relations x2i “ 1, xixj “ xjxi if |i ´ j| ě 2, pxixi`1q3 “ 1. We

can write this group as a Dynkin diagram by putting one dot per variable anda line between two dots if the corresponding variables satisfy pxixjq2 “ 1.

‚ ‚ ‚ ‚ ‚

This is a Dynkin diagram of type An, which corresponds to the Coxeter groupSn.

Problem 25. The cyclic group Cmn is isomorphic to Cm ˆ Cn if and only ifgcdpm,nq “ 1.

Here’s a really cool theorem about finitely generated abelian groups.

Theorem 26. Let A be a finitely generated abelian group. Then

A – Zr ˆ Z{d1Zˆ Z{d2Zˆ ¨ ¨ ¨ ˆ Z{dmZ

uniquely, such that di | di`1 for 1 ď i ă n.

We call the category of abelian groups Ab. For any subgroupB of an abeliangroup A, we have a quotient group A{B (for arbitrary groups, it only makessense for normal subgroups).

Okay let’s finally talk about group actions. An action of a group G on anobject X in some category C can be defined in several ways, but the followingis the cleanest.

Definition 27. A group action of a group G on X P C, where C is a category,is a homomorphism φ : GÑ AutpXq. We often write this G

œ

X . If X “ V is avector space, then φ is a representation.

Here’s an equivalent and more familiar definition:

Definition 28. A group action of a group G on X is a map G ˆ X Ñ X suchthat 1 ¨ x “ x for all x P X and g1pg2xq “ pg1g2qx.

8


Today we’re going to talk a little bit more about categories before we move onto G-modules.

Definition 29. Given two categories C and D, a (covariant) functor is a mapF : C Ñ D that assigns to each object c P C an object F pcq P D, and to each ar-row f : cÑ c1 in C, an arrow F pfq : F pcq Ñ F pc1q in D, subject to the followingconditions:

1. F pidcq “ idF pcq;

2. F pf ˝ gq “ F pfq ˝ F pgq.

A contravariant functor reverses the order of the arrows with composition,i.e. F pf ˝gq “ F pgq˝F pfq. The basic example of a functor is a forgetful functorthat forgets the structure of some object and regards it as a more basic one. Forexample, you can forget that a vector space is a vector space and think of it asa set. In the diagram below, each functor is forgetful.

Sets

k-Vect Ab Groups

There are analogues to injective and surjective for functors too.

Definition 30. A functor F : C Ñ D is faithful if the map F˚ : HomCpX,Y q Ñ

HomDpF pXq, F pY qq is injective for each pair X,Y P C. It is fully faithful if F˚is bijective for each X,Y P C.

These are not synonyms for injective and surjective! A faithful functor neednot be injective on objects or arrows. In the diagram above, the map Ab Ñ

Groups is fully faithful.Okay, that was your moment of categorical zen. Now let’s do some repre-

sentation theory.

Example 31. Let’s find the character table for Z{3Z. We first have the trivialrepresentation, and then we have the rotations by 2π{3 as another representa-tion. What’s the third one? It’s an abelian group, so has three conjugacy classesand therefore three irreducible representations.

Try as hard as we can, we can’t find the final representation over the realnumbers. We have to move to the complex numbers. Let ω “ e2πi{3. Thecharacter table for Z{3Z is below.

Z{3Z t0u t1u t2u

trivial 1 1 1

χ1 1 ω ω2

χ2 1 ω2 ω

9

Wait. . . aren’t the rows supposed to be orthogonal? They’re not if we usethe regular inner product. But they are if we use the Hermitian inner product

x~x, ~yy :“nÿ

i“1

xiyi.

Caveat: If a character is thought of as a one-dimensional representation,then it is multiplicative, χpghq “ χpgqχphq. But if χ “ trpρq for a higher-dimensional representation, then it is not multiplicative!

Now we return to our regularly scheduled program of group actions. Let’spick a group action φ : GÑ AutpSq.

Definition 32. 1. The orbit of an element s P S is tφpgqpsq | g P Gu;

2. the stabilizer of an element s P S is th P G | φphqpsq “ su;

3. the action is transitive if there is only one orbit.

We write Hs for the stabilizer of s P S. Observe that |Hs||S| “ |G|.

Example 33. The left multiplication action is φ : G Ñ AutSetspGq given byφpgqphq “ gh. This is an example of a transitive action. Note that we thinkof AutpGq as the automorphisms of G as a set, not as a group. Notice thatφpgqph1h2q “ gh1h2 but φpgqph1qφpgqph2q “ gh1gh2, so φpgq is not an automor-phism for each g.

Example 34. The conjugation action isψ : GÑ AutGroupspGq given byψpgqphq “ghg´1. This differs from the previous example because here, ψpgq is an auto-morphism for each g, because

ψpgqpxqψpgqpyq “ gxg´1gyg´1 “ gxyg´1 “ ψpgqpxyq.

The orbits of this action are the conjugacy classes of G.Additionally, conjugation acts on the set of subgroups ofG, by a new action

ψ1 : GÑ AutptH | H ď Guq defined as ψ1pgqpHq “ gHg´1.

Definition 35. H ď G is normal if H is stabilized by G under the conjugationaction. We write H CG if H is normal in G.

A normal subgroup is a union of conjugacy classes.

Example 36. There is a copy of the Klein four-group inside S4 that is generatedby p1 2qp3 4q, p1 3qp2 4q, p1 4qp2 3q, and it is normal. This is unique to thesymmetric group S4. For any other symmetric group Sn for n ‰ 4, the onlynormal subgroups are An, Sn and t1u.

Proposition 37. H C G is normal if and only if G{H is a group. Conversely,given a map φ : GÑ G is a homomorphism then kerφ ď G is normal.

10

Now we are equipped to talk about the lifting problem. Given V a vectorspace and U a subspace, we want to find a subspace W of V such that W isisomorphic to V {U .

V V {U

W


Let G be a group.

Definition 38. AG-module is a vector space V together with a homomorphismφ : GÑ GLpV q.

That is, a G-module is a representation of G. To write things concisely, wesometimes write G

œ

V . Additionally, we write the action of G as g ¨ ~v ratherthat φpgqp~vq.

Definition 39. In the category ofG-modules, which we callG-Mod, the objectsareG-modulesG

œ

V and the morphisms are the intertwiners f : V ÑW suchthat fpg~vq “ gfp~vq. We call this property G-linearity.

A cute remark (taking what Tom did and putting it into weird notation toobscure it) is that there is a forgetful functor F : G-Mod Ñ C-Vect which for-gets the action of G and forgets that the morphisms are G-linear. Essentially,this is saying that there is a subset of the vector-space homomorphisms be-tween V,W P C-Vect that is G-linear.

HomG-ModpV,W q Ă HomC-VectpV,W q.

Lemma 40 (Schur’s Lemma). If V,W are irreducible G-modules, then

(a) HomG-ModpV,W q “ 0 if V flW ;

(b) HomG-ModpV,W q – C if V –W .

Example 41. S3

œ

C3 by permuting the basis vectors, and S3

œ

C by the trivialrepresentation. Let f : C3 Ñ C be the function

fpv1e1 ` v2e2 ` v3e3q “ v1 ` v2 ` v3.

This is a C-linear map, and in addition it’s S3-linear. Take a look at the kernelof f .

ker f “ t~v P C3 | v1 ` v2 ` v3 “ 0u.

11

A basis for ker f is xe1 ´ e2, e2 ´ e3y. This space inherits an action of S3 by wayof its action on C3, and in fact this representation is irreducible.

C3 C

ker f

S3

fS3

What’s the matrix for the permutation p1 2q? Check it’s action on the basisvectors.

p1 2q ¨ pe1 ´ e2q “ e2 ´ e1 “ ´pe1 ´ e2q

p1 2q ¨ pe2 ´ e3q “ e1 ´ e3 “ pe1 ´ e2q ` pe2 ´ e3q

So the matrix is given by

p1 2q “

„

´1 1

0 1

Additionally, we can lift this representation to a representation on C3.

Problem 42. Find all the matrices for the action of S3 on ker f . Show thatthis representation is isomorphic to the other realization of the 2-dimensionalirreducible representation of S3 we found earlier.

Example 43. Another representation is pC,`q Ñ GL2C given by

z ÞÑ

„

1 z

0 1

.

This is a very important example! Similar to the Heisenberg group. The onlystable line is the line y “ x.

Theorem 44 (Maschke’s Theorem). Every representation of a finite group overR or C is a direct sum of irreducible subrepresentations. Each representationdecomposes into irreducibles.

A more sophisticated version says that kG is semisimple so long as the char-acteristic of k and order of G are relatively prime.

Proof. Given U Ă V invariant under G, we need to find a complimentary G-invariant subspace W Ă V such that U ‘W “ V . That is, we need to solvethe lifting problem of finding a subspace of V that is isomorphic to V {U . Thiswill be an orthogonal compliment, but not with respect to the usual Hermitianinner product on Cn.

Let’s first talk about what inner product we’re going to use here. The Her-mitian inner product is

~v ¨ ~w “

nÿ

i“1

viwi.

12

This inner product isn’t necessarily G-invariant, but we can fix that by averag-ing over all elements of the group G, as so

x~v, ~wy “1

|G|

ÿ

gPG

pg~vq ¨ pg ~wq.

Now we have a genuine G-invariant bilinear form on V .Then we take W “ UK, the orthogonal compliment of U inside V with

respect to our G-invariant bilinear form. We know that x~w, ~uy “ 0 for all ~w PW,~u P U . Claim that W is also G-invariant. Then, for any h P G,

0 “ x~u, ~wy “ xh~u, h~wy,

and we know that h~u P U sinceU isG-invariant, so now we know that h~w PW ,since it is perpendicular to anything in U . Hence, W is G-invariant as well.

This is really cool! It tells us that we can start with a representation V , pullout an irreducible component U , and then write V as V “ U ‘ UK. Now do itagain with UK, and eventually we will get down to a decomposition of V as

V “ U1 ‘ U2 ‘ . . .‘ Uk

into irreducible representations.

What happens when we take the direct sum of all of the irreducible repre-sentations of a group? We get something called the regular representation.

Definition 45. The regular representation of a groupG is the vector space withbasis xeg | g P Gy, with an action of G given by h ¨ eg “ ehg . This is also thegroup-algebra CrGs, and it has dimension equal to the order of G, dimCrGs “|G|. We can extend the action G

œ

CrGs linearly to an action CrGs

œ

CrGs.

Here’s a cool construction that shows that each irreducible representationappears at least once in CrGs. Given a decomposition into irreducibles,

CrGs “ U1 ‘ U2 ‘ . . .‘ Un,

and an irreducible representation G

œ

U . , choose ~u P U . Construct the func-tion f~u : CrGs Ñ U defined by f~upeidq “ ~u. Then for any g P G, f~upegq “f~upgeidq “ gf~upeidq “ g ¨ ~u.

Now we have a G-linear map f~u : CrGs Ñ U . This decomposes as a bunchof maps on each irreducible component fi : Ui Ñ U . By Schur’s Lemma, eachone of these maps is either an isomorphism or zero. But we cleverly con-structed f~u so that it was nonzero, so it must be the case that U – Ui for somei.

What about the converse? Now we want to know how many times eachirreducible representation appears. In particular, how often is U – Ui for eachi? The answer is that a irreducible representationU appears dimU -many times.

13

The proof of this is as follows. Let U be an irreducible representation. Con-sider HomGpCrGs, Uq. This space isomorphic to U as a vector space, with anisomorphism given by f ÞÑ fpeidq and its inverse ~u ÞÑ f~u. Then,

U – HompCrGs, Uq

– Hom

˜

nà

i“1

Ui , U

¸

–

nà

i“1

HompUi, Uq

Now by Schur’s lemma, HompUi, Uq – C if Ui – U , or zero otherwise. Andthese are isomorphic, so there must be dimU -many nonzero items on the right.This comes from a rather silly observation, that

U –nà

i“1

HompUi, Uq –dimUà

i“1

C.


Today, our groups are no longer finite (unless you maybe like to work overfinite fields). They’re going to be groups of matrices, LieGroups. Today we’regoing to talk about examples, and then for the next few days we’ll talk aboutwhat it really means to be a Lie group. We’ll talk about differential geometryand eventually, algebraic geometry, because the Lie groups we care about arealgebraic.

For now, all I have to do is convince you that Lie groups are interesting. Fornow, let’s let this ad-hoc definition suffice:

Definition 46 (Temporary). A Lie group is a group of matrices defined by apolynomial condition on GLnk, where k is either R or C.

Example 47. The general linear group is GLnC “ t invertible nˆ n matrices u.This is contained in the set of all nˆ n matrices, glnC, and GLnC is the compli-ment of the locus where the determinant vanishes.

GLnC “ glnCztX P glnC | detX “ 0u

The same thing works over R instead of C.We have a multiplication map m : GLnCˆGLnCÑ GLnC given by

paijq , pbijq ÞÑ

˜

nÿ

k“1

aikbkj

¸

And there is an inversion map i : GLnCÑ GLnC given by Kramer’s rule. Bothm and i are polynomial maps in the entries of matrices A and B.

14

Example 48.

GL2R “

"„

a b

c d

ˇ

ˇ

ˇ

ˇ

ad´ bc “ 0

*

If you think about this really hard, you will see that this is like a cone. Becauseif ad´ bc “ 0, then so does λpad´ bcq “ 0 as well for any λ, and it’s singular atthe origin a “ b “ c “ d “ 0. We can think of it as a subset of R4 as well.

Example 49. The special linear group is a smooth submanifold of GLnC.

SLnC “ tA P GLnC | detpAq “ 1u

Example 50. The length-preserving transformations of Rn are the orthogonalmatrices.

On “ tA P GLnR | A~v ¨A~w “ ~v ¨ ~w for all ~v, ~w P Rnu

We can reinterpret the length-preserving condition as ATA “ I , because

~vT ~w “ ~v ¨ ~w “ A~v ¨A~w “ pA~vqTA~w “ ~vT pATAq~w

Choosing ~v “ ~ei and ~w “ ~ej , we see that pATAqij “ δij for all i.The rows and columns of orthogonal matrices are of unit length and or-

thogonal.

Example 51. The length-preserving transformations of Cn are the unitary ma-trices.

Un “ tA P GLnC | xA~v,A~wy “ x~v, ~wy for all ~v, ~w P Cnu.

By the same argument as in the previous example, we see that unitary matricesmust satisfy A˚A “ I .

The rows and columns of unitary matrices are of unit length and orthogonalwith respect to the Hermitian inner product.

Fun fact: Un is not algebraic because of the complex conjugates.

Here are some interesting inclusions of Lie groups.

GLnC Ą Un

SLnC Ą SUn

SOnC Ą SOnR

Along with the symplectic group, Sp2nC, these are all of the Lie groups withonly five exceptions. Let’s do some examples.

Example 52. In dimension 1,

GLp1,Cq “ C˚ Ą Up1q “ unit circle

SLp1,Cq “ 1

Op1,Cq “ ˘1

15

In dimension n, we have a map det : Upnq Ñ Up1q that takes a unitarymatrix to some value on the unit circle.

In dimension 2, we have

SOp2,Rq “"„

cos θ ´ sin θ

sin θ cos θ

*

SOp2,Cq “"„

a ´b

b a

ˇ

ˇ

ˇ

ˇ

a2 ` b2 “ 1, a, b P C*

“

"„

z ` z´1{2 ´pz ´ z´1{2iq

´pz ´ z´1{2iq z ` z´1{2

*

– C˚

We should talk about SUp2q at the same time we talk about SOp3,Rq. As therotations in R3, SOp3,Rq is the rotational symmetries of the two-sphere S2.

Fun fact: if A P SOp3,Rq then A has an eigenvector with eigenvalue 1. Thisis because the characteristic polynomial looks like

x3 ´ trpAqx2 ` bx´ detpAq,

and detpAq “ 1. This polynomial has three roots, and one of them must be realbecause all of the coefficients are real. These roots multiply to 1, so the real rootmust be ˘1.

Once we know that there is an eigenvector with eigenvalue 1, this corre-sponds to a direction that A leaves unchanged – the axis of rotation.

Now let’s talk about SUp2q. These matrices look like„

z ´w

w z

such that zz ` ww “ 1. This is exactly the formula for the 3-sphere S3.We can rewrite the definition of an element of SUp2q by setting z “ a ` bi,

w “ c` di with a, b, c, d P R. Then,„

a` bi ´c` di

c` di a´ bi

“ a

„

1 0

0 1

` b

„

i 0

0 ´i

` c

„

0 ´1

1 0

` d

„

0 1

1 0

Call this a~1` b~i` c~j ` d~k. Then the multiplication rules for these matrices are

~i2 “ ~j2 “ ~k1 “~i~j~k “ ~1

We got the unit quaternions!

H “ ta` b~i` c~j ` d~k | a2 ` b2 ` c2 ` d2 “ 1u

This is the equation for a sphere in R4, so SUp2q is topologically the 3-sphereS3.

Another fun fact. On the sphere S3 there is an action of SUp2q for which thelatitudes of SUp2q are conjugacy classes and the longitudes are stabilizers!

SUp2q is a double cover for SOp3,Rq as a manifold.

16


A Lie group is a group in the category of smooth manifolds. That is the state-ment that we’re trying to understand this week. So far, all we know is thatthey’re sitting inside the Euclidean space Cn2

and are described by polyno-mial equations on the coordinates. For instance, SLpn,Cq is described by theequation detpAq “ 1.

Not Compact Compact(Uncool) (Cool)SLpn,Cq SUpnqSOpn,Cq SOpn,Rq

From here we’re going to try and understand these things as manifolds. Todaywe’re going to do some topology. Friday will be function orgy day.

Definition 53. A topology on a setX is a collection of subsets U Ď X such that

(i) H, X are open;

(ii) U1, . . . , Un open implies that U1 X . . .X Un is open;

(iii) tUi | i P Iu each open impliesŤ

iPI Ui is open.

Example 54. The Euclidean topology on Rn (or Cn » R2n) has open sets U thatare unions of open balls Brppq “ tx P Rn | }x´ p} ă ru. That is, every open setU is of the form

U “ď

iPI

Brippiq.

Remark 55. I’m going to say “balls” a lot. I’ll try not to giggle too much. Sorryabout that.

Example 56. Let X be the real line with two origins, pictured below. This canbe described as RY t01u, where 01 is a second origin. Open sets are open inter-vals in R and any open set containing 0 intersects every open set containing 01.It looks like this:

This space does not have a topology coming from a metric, and it is non-Hausdorff.

We want a category of topological spaces, so we should define maps be-tween them.

Definition 57. A map f : X Ñ Y of topological spaces is continuous if andonly if U Ă Y open implies f´1pUq Ă X is open.

Definition 58. A homeomorphism between topological spaces is a continuousfunction with a continuous inverse.

17

Example 59. Polynomials P px1, . . . , xnq P Rrx1, . . . , xns are continuous func-tions P : Rn Ñ R with respect to the Euclidean topology.

If Y is a subset ofX , andX is a topological space, we can define an inducedtopology on Y by stating that the open sets of Y are exactly pU XY q for U openinX . This has the following universal property. Let i : Y Ñ X be the inclusion.The topology on Y is defined by stating that for any function f : Z Ñ Y iscontinuous if and only if i ˝ f : Z Ñ X is continuous.

X

Z Y

i˝f

f

i

This fact and the previous example allows us to conclude that all of our ex-amples of Lie groups are topological spaces: they are subspaces of Cn2

definedby polynomial equations, which are continuous.

Note that Rn fi Rm for n ‰ m.

Definition 60. A topological space X is

• Hausdorff if and only if for all x1, x2 P X , there are U1, U2 open such thatx1 P U1, x2 P U2 yet U1 X U2 “ H;

• disconnected if there are nonempty, distinct U1, U2 open in X such thatU1 X U2 “ X yet U1 X U2 “ H;

• compact if for all open covers X “Ť

αPA Uα, there is a finite subcoverX “

Ťni“1 Uαi ;

• a manifold if it is Hausdorff, locally homeomorphic to Euclidean space,and there is a countable cover of basic open sets.

We require that the intersections of charts on a manifold are compatible.

Theorem 61 ((Heine-Borel)). A subspace X Ă Rn with the induced Euclideantopology is compact if and only if X is closed and bounded.

Example 62. Let’s think about SOpn,Rq as a subset of Rn2

. Each matrix inSOpn,Rq is of the form

A “

»

–

| | |

~u1 ~u2 . . . ~un| | |

fi

fl ,

which is a vector in Rn2

. Each of the columns is orthogonal to each other, soit looks like going out in some direction, and then taking a right angle turn in

18

another direction, and so on. Since each vector is length one, then the furthestdistance we travel from the origin is

?n.

Additionally, SOpn,Rq is closed because it is the inverse image of a closedset under bunch of polynomial equations:

~ui ¨ ~ui “ 1 ~ui ¨ ~uj “ 0 detpAq “ 0.

So we conclude by Heine-Borel that SOpn,Rq is compact.

Example 63. An example of non-compact Lie group is SLp2,Rq Ă R4. It con-tains an unbounded line (well, excluding the origin)

t ÞÑ

„

t 0

0 1t

.

Here’s an example of gluing. Let U1 “ Rn and let U2 “ Rn. Let U12 “

Rnzt0u and let U21 “ Rnzt0u. Glue by the identity id “ f12 : Rnzt0u Ñ Rnzt0uto get an n-space with two origins. This is not Hausdorff. Every open setcontaining one origin intersects every open set containing the other.

If instead we glue along the map

f12p~vq “~v

}~v}2,

then we get the sphere. One origin becomes a point at8 and we get somethinghomeomorphic to Sn in Rn`1. This is the stereographic projection.


Today we’re talking about Young Diagrams in a box. The conjugacy classes ofthe symmetric group are indexed by Young diagrams. For example, in S4 wehave

.

But this is too disorganized for me, so I want to put them in a box. For S4,we’re going to put them in a 2ˆ2 box and call itGp2, 4q. The elements ofGp2, 4qare

H

Or we could have Gp2, 5q, in which we’re trying to fit them in a box of height2 and width 5 ´ 2 “ 3. How many Young diagrams can we fit in a box of thisshape?

H

19

Notice that these have much more symmetry than the usual Young dia-grams. There are dual diagrams for each diagram, which can be found byfilling in the rest of the squares in the box and then rotating by π. For example,in a 3ˆ 3 box, we find the dual by

ù

X X XX ù X X

X X Xù

We also define a partial ordering on the Young diagrams by saying that onediagram is less than another if we can obtain the second by adding a singlebox. For example, in Gp2, 4q, we have

H

These diagrams are how we will describe some Grassmanians, but first weshould talk about projective space.

Definition 64. Projective space PnC is the set of lines through the origin inCn`1. Namely,

PnC “ tpx0 : x1 : . . . : xnq P PnC | xi ‰ 0 for some iu,

where px0 : x1 : . . . : xnqmeans we consider points up to (nonzero) scalar mul-tiplication.

We can obtain projective space by gluing two copies of C. On the projectiveline P1C, we have coordinates

px0 : x1q “

#

px0{x1, 1q x1 ‰ 0

p1, x1{x0q x0 ‰ 0

We can make the two cases into two copies of the complex numbers. Call themU0 and U1, where U0 is the subset of P1C where x0 ‰ 0 and U1 is the subsetwhere x1 ‰ 0.

U0 “ tpx0 : x1q | x0 ‰ 0u “ tp1: zq | z P Cu

20

U1 “ tpx0 : x1q | x1 ‰ 0u “ tpz : 1q | z P Cu

We glue along the function f : U0ztp1: 0qu Ñ U1 given by

fp1, zq “ p1{z, 1q.

It is well-defined because we omit the only possible bad point p1: 0q. Then,we identify a point in U1 with its image under f . This gives us CP1 as twooverlapping copies of C, each missing a point that the other contains. This lastpoint which is not in the domain of f is the point at infinity.

We can also do this in general, gluing copies of the affine plane to get pro-jective space.

In P2C, thinking about how to get all the lines through the origin in thisway gives us a distinguished flag of subspaces

0 Ă xe0y Ă xe0, e1y Ă xe0, e1, e2y;

this is a point, and a line containing that point, and then a plane containing theline, and finally a 3-space.

We get this from first setting z “ 1, and from this we get a bunch of lines butmiss all those in the xy-plane because they don’t pass through the plane z “ 1

in C3. This corresponds to the subspace xe0, e1, e2y; this is lines in xe0, e1, e2y yetnot in xe0, e1y. Then by setting y “ 1, we add all of the lines through the originexcept the line x “ z “ 0. This corresponds to the portion of the flag xe0, e1y;this is lines in xe0, e1yzxe0y. Finally, we set x “ 1 and get the final subspace oflines in xe0y.

The generic behavior is

Lines Subspace Coordinates` R xe0, e1y C2 z “ 1

` P xe0, e1y; ` R xe0y C y “ 1

` P xe0y C x “ 1

Definition 65. A flag is a sequence of vector spaces increasing in dimension

V0 Ď V1 Ď . . . Ď Vn,

and a full flag for a vector space V is a sequence of spaces that increase indimension by one each time, and end up filling the whole space. For instance,if V has dimension n, then a full flag is

0 “ V0 Ď V1 Ď . . . Ď Vn “ V,

and Vi has dimension i.

A generalization of flags are the Grassmanians.

21

Definition 66. The Grassmanian Gpm,nq is the set of subspaces of Cn of di-mension m.

Gpm,nq “ tW Ď Cn | dimW “ mu.

Much like projective space, we want to give coordinates to points in theGrassmanian. We will do this by describing the points with m coordinates. Ifthe coordinates of Cn are x1, . . . , xn, then a point is described by

pxi1 , . . . , ximq

for an index set I “ p1 ď i1 ă i2 ă . . . ă im ď nq, each 1 ď ij ď n. In projectivespace we set one coordinate to 1, but in a Grassmanian we set m coordinates to1.

There is a much more sane way to describe these coordinates, much like theratios px0 : . . . : xnq in PnC.

Definition 67. The Grassmann coordinates for a point in Gpm,nq is an mˆ nmatrix

X “

»

—

–

x11 x12 . . . x1n...

.... . .

...xm1 xm2 . . . xmn

fi

ffi

fl

where the basis of a space W is the rows of this matrix. We consider suchmatrices as equivalent up left-multiplication by GLpm,Cq, which only changesthe basis but doesn’t leave the space W .

Definition 68. From the Grassmann coordinates, we can define the PluckerCoordinates as points in P p

nmq´1 by taking determinants of minors

ˆ

detpMI1q : detpMI2q : . . . : detpMIpnmqq

˙

,

where Ii Ă t1, . . . , nu are all the distinct subsets of t1, . . . , nu and MI is theminor of the Grassmann coordinate matrix X indexed by I .

This also gives us the Plucker embedding into Ppnmq´1.

Example 69. For Gp2, 4q, the Plucker„

x11 x12 1 0

x21 x22 0 1

PluckerÝÝÝÝÑ px11x22 ´ x12x21 : ´ x21 : x11 : x12 : 1q

Because all of the Grassmann coordinates appear, this map is injective!


The Grassmannian – Done Right

Recall thatGrpm,nq “ tW Ă Cn | dimpW q “ nu.

22

We also have a Plucker embedding

Grpm,nq ãÑ Ppnmq´1C.

Today we’re going to learn about the Schubert cells which stratify the Grass-mannian. The Schubert cells are indexed by Young diagrams in a box.

Definition 70. A Young diagram in a box is

λ “ tn´m ě λ1 ě λ2 ě . . . ě λm ě 0u

For one of these Young diagrams in a box we have m rows and n´m columns,and the diagrams look like

m rows"

X X XX X XX X

looooomooooon

n´m columns

Problem 71. There are`

nm

˘

such λ.

We also say that

|λ| “ λ1 ` . . .` λm ď mpn´mq,

and have a compliment of a Young diagram in a box given by

λc “ tn´m ě pn´mq ´ λm ě . . . ě pn´mq ě λ1 ě 0u

We have a partial order on these Young diagrams in boxes given by λ ď µ ifand only if µ “ λ` some squares .

To begin with these Schubert cells, we fix a full flag

0 Ă xe1y Ă xe1, e2y Ă . . . Ă xe1, . . . , en´1y Ă Cn

For ease of notation, call it

0 “ V0 Ă V1 Ă V2 Ă . . . Ă Vn´1 Ă Vn “ Cn

What is the generic behavior of a W Ă Cn of dimension m with respect to thisflag? We might have

dimpWXV1q “ 0, . . . ,dimpWXVn´mq “ 0,dimpWXVn´m`1q “ 1, . . .dimpWXVnq “ m.

Generically, we get something like

dimpW X Vn´m`iq “ i,

23

and the dimension sequence of W X Vi looks like

p0, 0, . . . , 0loooomoooon

n´m

, 1, 2, 3, . . . ,mq.

We can also look at the jump sequence of successive differences in dimension.For the above, this is

p0, 0, . . . , 0loooomoooon

n´m

, 1, 1, 1, . . . , 1q.

By some linear algebra shenanigans, observe that the vectors of V which lie inW X Vn´m`1 look like

p˚, ˚, . . . , ˚loooomoooon

n´m

, 1, 0, 0 . . . , 0q

with respect to the basis that comes from our flag. And likewise, we can forcethe vectors of V which lie in W X Vn´m`2 to look like

p˚, ˚, . . . , ˚loooomoooon

n´m

, 0, 1, 0 . . . , 0q.

Putting these vectors into a matrix, we get the Grassmann coordinates! Apoint in W looks like

»

—

—

—

–

˚ . . . ˚ 1 0 . . . 0

˚ . . . ˚ 0 1 . . . 0...

. . ....

......

. . ....

˚ . . . ˚ 0 0 . . . 1

fi

ffi

ffi

ffi

fl

.

Now what about a non-generic subspace W ? This means that the sequenceof jumps looks like

p0, . . . , 0, 1, 0, . . . , 0, 1, 0, . . . , 0, 1, . . .q.

If W has dimension m, there are m-many ones in the above sequence – thedimension of intersections increases by 1 a total of m times.

Example 72. What about Grp2, 4q? We calculate with respect to the standardflag

0 Ă xe1y Ă xe1, e2y Ă xe1, e2, e3y Ă C4.

The jump sequence p0, 0, 1, 1q is the most generic, and corresponds to the matrix„

˚ ˚ 1 0

˚ ˚ 0 1

.

The jump sequence p0, 1, 0, 1q is not quite so generic, and corresponds to thematrix

„

˚ 1 0 0

˚ 0 ˚ 1

.

24

The jump sequences are themselves indexed by Young diagrams, and there isa partial order to them

1001

0011 0101 1010 1100

0110

Look familiar? The jump sequences correspond to Young diagrams as fol-lows. The most generic sequence corresponds to the empty Young diagram.Then for any other jump sequence, observe how many positions the left 1 hadto move to get to it’s position, and that is the first row of a Young diagram.The number of positions that the rightmost 1 moved to it’s new position is thenext row, and so on. The biggest Young diagrams correspond to the smallestsubspaces in the stratification of the Grassmannian. So for example,

Example 73. The jump sequence p0, 0, 1, 1q corresponds to an empty Young di-agram. To get the jump sequence p0, 1, 0, 1q, the leftmost 1 moved one position,so we get

p0, 1, 0, 1q ù

To get the jump sequence p1, 0, 0, 1q, we moved the leftmost 1 by two positions,so

p1, 0, 0, 1q ù

The rest of them are as follows

p0, 1, 1, 0q ù

p1, 0, 1, 0q ù

p1, 1, 0, 0q ù

Another example. The jump sequence p1, 0, 1, 0q corresponds to„

1 0 0 0

0 ˚ 1 0

.

Definition 74. Given a full flag F and a Young diagram in a box λ, the Schu-bert cell is

SFλ “

"

s subspaces W Ă Cn correspondingto λ with respect to the flag F

*

25

We won’t prove this following result, although we will give a plausibilityargument. The closure of Sλ in the Grassmannian is

Sλ “ď

λďµ

Sµ

What happens if we reverse the flag? If we take the flag

0 Ă xe1y Ă xe1, e2y Ă . . . Ă xe1, . . . , en´1y Ă Cn

and substitute the flag

0 Ă xeny Ă xen, en´1y Ă . . . Ă xen, . . . , e1y Ă Cn,

what happens to the Schubert cells? You switch the rows and reverse them.Let’s illustrate by example:

Example 75. In Grp2, 4q,„

1 0 0 0

0 ˚ ˚ 1

ù

„

1 ˚ ˚ 0

0 0 0 1

These complimentary Schubert cells intersect in a point! This is Poincareduality.


This week, we’ll talk about some algebraic geometry related to the Lie groupsthat were introduced last week. The Lie groups we care about are

SLpn,Cq Ą SUpnq

SOpn,Cq Ą SOpn,Rq

Of these, SLpn,Cq is “algebraic,” SUpnq, SOpn,Cq and SOpn,Rq are compact.We also care the spaces

PnC, Grpm,nq, Flpnq,

which are both algebraic and compact. The rightmost one above is the flagvariety.

As Tom Garrity said in his first lecture, “functions describe the world.” Thisis very true of algebraic geometry, where everything eventually gets replacedwith functions rather than objects.

For instance, we can realize this slogan in the group algebra CrGs. All rep-resentations of a finite group G are contained inside CrGs, which we can thinkof as functions on G, with multiplication convolution.

CrGs “ tf : GÑ Cu “

#

ÿ

gPG

fpgqg

+

.

26

Let M be a manifold. Because M is a topological space, we know what itmeans for function f : M Ñ R to be continuous. But what does it mean for fto be differentiable? We can define this in terms of the functions on M .

Remark 76. From an action G

œ

M of a group on a manifold, we get an actionof G on the functions f : M Ñ R defined by

g ¨ fpxq “ fpg´1xq.

Hierarchy of Functions

There is a hierarchy of functions f : RÑ R

(1) continuous functions, for which limhÑ0

|fpx0 ` hq ´ fpx0q| “ 0;

(2) differentiable functions, for which

limhÑ0

|fpx0 ` hq ´ y1h´ fpx0q|

|h|“ 0,

for some y1, which we call f 1px0q;

(3) smooth (infinitely differentiable) functions, for which f 1pxq, f2pxq, etc.are differentiable as functions of x.

Smooth functions have Taylor series,

fpx0 ` hq « y0 ` y1h` y2h2

2` . . .

where yi “ f piqpx0q. Of course, there are functions like fpxq “ e´1{x2

that don’tagree with their Taylor series, so we add a few new class to our hierarchy.

(4) Analytic functions, which agree with their taylor series everywhere;

(5) rational functions, which are ratios of polynomials fpxq “ ppxq{qpxq forp, q P Rrxs.

Over the complex numbers, the hierarchy goes like this.

(1) Continuous functions;

(2) holomorphic functions (which are complex-differentiable and smoothand analytic)

limhÑ0

|fpz0 ` hq ´ y1z0 ´ y0|

|h|“ 0,

where we set f 1pz0q “ y1 and set y0 “ fpz0q.

27

So over the complex numbers, analytic, differentiable and smooth are thesame notions. Holomorphic functions satisfy the maximum principle.

Proposition 77 (Maximum principle). If f : C Ñ C is holomorphic in a con-nected open set U Ď C and z0 P U such that |fpz0q| ě |fpzq| for all z in aneighborhood of z0, then f is constant on D.

We can also talk about the hierarchy for functions in several variables f : Rn ÑR.

Definition 78. A function f : Rn Ñ R is differentiable at a point ~x0 P Rn if

lim~hÑ0

|fp~x0 ` ~hq ´ ~y1 ¨ ~h´ ~y0|

|~h|“ 0.

The derivative is the gradient

∇fp~x0q “ ~y1 “

ˆ

Bf

Bx1p~x0q, . . . ,

Bf

Bxnp~x0q

˙

.

We can also talk about infinitely differentiable and analytic functions.What about multivariable functions over C? A function f : Cn Ñ C is holo-

morphic if and only if it is holomorphic in each variable separately. This fact isHartog’s theorem.

The Implicit function theorem tells us that if we have a vector-valued func-tion ~f : Rn Ñ Rm and write ~f “ pf1, . . . , fmq, we can rewrite some portion ofthe image of ~f as the zero set of the functions f1, . . . , fm. We can reinterpretthis theorem as follows.

Theorem 79 (Implicit Function Theorem). If ~f : Rn Ñ Rm or ~f : Cn Ñ Cm,then

Vp~fq “ t~x P Cm | ~fp~xq “ 0u

is an analytic manifold near any point ~x0 for which

lim~hÑ0

|fp~x0 ` ~hq ´ Jacp~fqp~x0q~h´ ~fp~x0q|

|~h|“ 0,

and the Jacobian Jacp~fq has full rank near ~x0.

Example 80. The circle fpx, yq “ x2` y2´ 1 has gradient∇f “ p2x, 2yq, whichhas rank 1 at the point p0, 1q. So we can rewrite the y-coordinate in terms of xnear p0, 1q, as y “

?1´ x2.

28

Example 81. We can use this to analyze the Lie groups from before. Is SLpn,Cqa manifold? Well, elements X of SLpn,Cq are zeros of the polynomial fpXq “detpXq ´ 1. So yes, it is a manifold. The partial derivatives of this equationcan be found by expanding the determinant along minors Mij correspondingto removing the i-th row and j-th column.

Bf

Bxij“ p´1qi`j detpMijq

Example 82. What about Opn,Cq? This seems like a difficult Lie group to ana-lyze, because the polynomial equation is a bit more complicated here. Orthog-onal matrices A are zeros of fpAq “ ATA ´ I . We can expand this around theidentity for A “ I ` εB, where ε is small. On one hand,

fpI ` εBq “ fpIq ` JacpfqεB Òpε2q

but on the other hand,

fpI ` εBq “ pI ` εBqT pI ` εBq ´ I “ 0` εpBT `Bq ` ε2BTB Òpε3q,

so we conclude that Jacpfq is the map B ÞÑ B `BT .This shows that Opn,Cq is a manifold, but we also know now how to com-

pute the tangent space at the identity: it’s the kernel of the Jacobian operator.

kerpJacpfqq “ tB | B `BT “ 0u “ tSkew-symmetric matricesu.

Example 83. We can analyze SLpn,Cq in the same way. Our polynomial isfpAq “ detpAq ´ 1, and near the identity

fpI ` εBq « detpI ` εBq ´ 1` εtrpBq Òpε2q,

so Jacpfq “ tr, and the tangent space is

slpn,Cq “ t traceless nˆ n matrices u

29


Tensor Madness

Between Tom Garrity and Sean McAfee, we’ve already covered most of what Iwant to say about tensors. So we’ll fly through it really quickly.

Definition 84. Let V be a finite dimensional vector space over C. The dualspace V ˚ is

V ˚ “ t linear f : V Ñ Cu.

If G

œ

V , then we also get an action G

œ

V ˚ by g ¨ fpxq “ fpg´1 ¨ xq.

Definition 85. The tensor algebra of a finite-dimensional vector space V is thespace

TV “

8à

n“0

V bn “ C‘ V ‘ pV b V q ‘ . . .

IfG

œ

V , then we also have an actionG

œ

TV by acting on the componentsof a tensor individually. For the d-th tensor power V bd, we have a right-actionof Sd by permuting the coordinates. The action of Sd commutes with the actionof G.

Joke 86. “I refuse to do it any other way, even if Serre tells me I have to. I knowSerre. I don’t know if he knows me, but . . . ” - Aaron Bertram.

We can break up V bd into irreducible components under the action of Sd.One component is the symmetric tensors, SymdV , and another component isŹd

V , the antisymmetric tensors. We have a decomposition

V bd “ SdV ‘ľd

V ‘ p other stuff q .

Example 87. For example, Sym2V is the tensors that look like xb x, yb y, andxb y ` y b x, and

Ź2V is the tensors that looks like xb y ´ y b x.

Definition 88. The symmetric algebra over a finite-dimensional vector spaceV is

SympV q “8à

n“0

SymnV “ C‘ V ‘ Sym2V ‘ . . .

This is just the polynomial ring Crx1, . . . , xdimV s.

Definition 89. The exterior algebra over a finite-dimensional vector space Vis

ľ

V “8à

n“0

ľnV “ C‘ V ‘

ľ2V ‘ . . .

30

Notice that for n ě dimV ,Źn

V “ 0. Hence, the exterior algebra remainsfinite dimensional even though the symmetric algebra is not finite dimensional.Additionally, det P

ŹnV ˚.

Definition 90. The super polynomial algebra over V is

8à

dě2

´

SymdV ‘ľd

V¯

The super-polynomial algebra is useful in physics.We can also relate this to invariant theory once we choose a basis e1, . . . , en

for V , through the symmetric algebra.

SympV q “ Crx1, . . . , xns“ Sym‘Anti-Sym‘ . . .

“ Cre1, . . . , ens ‘∆ ¨ Cre1, . . . , ens ‘ . . . ,

where ∆ is the polynomial that’s invariant under the action of An,

∆ “ź

iăj

pxi ´ xjq.

Definition 91. For a partition λ “ pλ1, λ2, . . . , λnq, let

Pλ “ xλ1`pn´1q1 ¨ ¨ ¨x

λn´1`1n´1 xλnn ` . . .` sgnpσqxλ1`pn´1q

σp1q ¨ ¨ ¨xλn´1`1

σpn´1q xλnσpnq,

where the sum is over all permutations σ P Sn. Then the Schur Polynomial isPλ{P0, and P0 “ ∆.

Algebraic Geometry

To motivate this, let’s think about the continuous characters of Up1q. These aremaps χ : Up1q Ñ Cˆ, and χpeiθeiψq “ χpeiθqχpeiψq. This tells us that χpeiθqmust lie on the unit circle. Moreover, χ maps roots of unity to roots of unity.From this, we can conclude that the only continuous characters of Up1q aremaps χkpgq “ gk for k P Z.

So all of the representations of Up1q lie insideà

kPZCeikθ

We can account for these characters in another way, by considering Up1q ĂCˆ “ GLp1,Cq. What are the algebraic functions on Cˆ “ GLp1,Cq? These arethe Laurent polynomials, because we don’t have to worry about plugging inzero in the denominator.

Crz, z´1s “à

kPZCzk

31

This looks a lot like what we found for Up1q!From this toy example, it makes sense to think about the algebraic func-

tions on other Lie Groups that are more complicated, like GLpn,Cq, Spp2n,Cq,SLpn,Cq, or SOpn,Cq. Given an algebraic group G, we want to write the coor-dinate ring of functions on the variety G as a sum of the irreps of the group G.This is what we did for Up1q above.

We’ll be able to forget the group entirely, and focus on just the functions onthe group.

Example 92. For the variety SLpn,Cq, the coordinate ring is

Crx11,x12,...,xnns{xdetpxijq´1y

When n “ 2, we get

Crx11,x12,x21,x22s{xx11x22´x21x12´1y.

The coordinate ring of GLpn,Cq is given by

Crtxiju,ys{xy detpxijq´1y “ C“

txiju,det´1‰

We can’t write down a polynomial equation for GLpn,Cq in n2 dimensions,because it’s the compliment of the zeros of a polynomial equation. Namely, it’sthe compliment of detpxijq “ 0. But we can do it in n2`1 dimensions as above.


Today we’re going to talk about the category of affine varieties. In particular,we’re going to focus on the ones that are finite-type over C.

Let’s start with an analogy to differential geometry. Let M be a topologicalmanifold. We have a sheaf CM of continuous functions on M . For any open setU ĎM ,

CM pUq “ t continuous f : U Ñ Ru.

There is another example of a sheaf on M which we call ZM . This is the sheafof integer-valued continuous functions on any open set,

ZM pUq “ t continuous f : U Ñ Zu.

Although this looks rather boring, (these f : U Ñ Z are constant on connectedcomponents of U ) it contains a lot of information. From it, we can find all ofthe Betti Numbers of the manifold.

If M is instead a smooth manifold, there is another sheaf contained withinCM called C8M . This is defined by

C8M pUq “ t smooth f : U Ñ Ru.

32

And there’s even a finer sheaf, which we call OM ,

OM pUq “ t analytic f : U Ñ Ru.

To summarize, we have the following inclusions

CM Ą C8M Ą OM Ą ZM .

In all of these cases, we have a topological spaceM with a sheaf of functionson it. This is what the picture will look like in algebraic geometry, but thetopological space will not be Hausdorff (among other failings).

Definition 93. A finitely-generated C-algebra is a C-vector space A equippedwith a multiplication operation of vectors such that there are a1, . . . , an P A sothat the map Crx1, . . . , xns Ñ A defined by xi ÞÑ ai is surjective.

In analogy to the differential geometry setting, we will have a topologicalspace SpecpAq and a sheaf of regular functions OXpUq.

There are two theorems of Hilbert that are critical to get us going in alge-braic geometry.

Theorem 94 (Hilbert’s Basis Theorem). Any ideal I C Crx1, . . . , xns is finitelygenerated.

Theorem 95 (Nullstellensatz). The maximal ideals in Crx1, . . . , xns are the ker-nels of evaluation maps at points ~a “ pa1, . . . , anq P Cn, i.e. the kernels of mapsev~a : Crx1, . . . , xns Ñ C defined by ev~apfq “ fp~aq.

kerpev~aq “ xx1 ´ a1, . . . , xn ´ any

Definition 96. The max-spectrum of a ring A is the set of all maximal ideals ofA.

maxSpecpRq “ tmaximal ideals of Au

Remark 97. LetA be a C-algebra with generators a1, . . . , an and map φ : Crx1, . . . , xns ÑA. Then if kerφ “ xf1, . . . , fmy, we know that

A –Crx1,...,xns{kerφ “

Crx1,...,xns{xf1,...,fmy,

so ideals ofA are in bijection with ideals of Crx1, . . . , xns containing f1, . . . , fm.We identify Crx1, . . . , xns{ kerφ with the variety Vpkerφq corresponding to theideal kerφ. In particular, the set of these fi can be thought of as polynomialsdefining the variety Vpkerφq, which has coordinate ring A. This gives a bijec-tion

maxSpecpAq ÐÑ Vpf1, . . . , fmq.

What’s the topology on our space?

33

Definition 98. The Zariski topology on maxSpecpAq is the topology with theclosed sets

ZpIq “ tm | I Ă mu Ă maxSpecpAq.

Joke 99. The Zariski topology is like Texas. There is nothing small in theZariski topology. Every open set is dense! All open sets meet each other.

Problem 100. Show that the Zariski topology forms a topology on maxSpecpAq.

In the Zariski topology, we have a basis of open sets given by

Uf “ maxSpecpAqzZpxfyq “ tm | f R mu

For an ideal I “ xf1, . . . , fmy, we have an open set

mď

i“1

Ufi “ maxSpecpAqzZpIq.

An application of the basis theorem says that if we have an increasing chainof ideals,

I1 Ď I2 Ď I3 Ď . . . Ď In Ď . . .

it must eventually stabilize, that is, there is some n such that for all m ě n,Im “ In. This tells us that we cannot have infinitely decreasing chains of closedsets. They must stabilize

ZpI1q Ě ZpI2q Ě . . . Ě ZpInq “ ZpIn`1q “ . . . “č

i

ZpIiq.

Definition 101. (a) A topological space in which every nested sequence ofclosed sets stabilizes is called Noetherian;

(b) a Noetherian topological spaceX is irreducible if for any closed Z1, Z2 Ď

X such that X “ Z1 Y Z2, we must have X “ Z1 or X “ Z2.

Fact 102. The Zariski topology on an integral domain is necessarily irreducible.

Example 103. Consider A “ Crx, ys{xxyy. This is not a domain, because x ‰ 0,y ‰ 0, yet xy “ 0. The maximal spectrum of A is the two coordinate axes:

Zpxxyyq “ Zpxxyq Y Zpxyyq.

Zpxyyq

Zpxxyq

34

The second thing that we needed in the differential geometry version ofthis was a sheaf of functions on the manifold. So here we’re going to define thesheaf of regular functions on a domain A.

For any domain A, we can form the field of fractions

CpAq “"

a1a2| a1, a2 P A, a2 ‰ 0

*

Regard elements a P A as functions on maxSpecpAq, defined by

apmq “ apmod mq P C.

The reason that these elements live in C is because A is the coordinate ring ofsome variety, and so the elements of A are functions on that variety. Hence, wecan imagine taking a polynomial f modulo a maximal ideal m as evaluation off at the point corresponding to m.

We call elements a P A regular functions on maxSpecpAq. Any elementφ “ a

b P CpAq also defines a function on maxSpecpAq as

φpmq “a

bpmq “

apmod mq

bpmod mq,

wherever b pmod mq ‰ 0. For each φ P CpAq, we write the domain of ofdefinition as those points where b pmod mq is nonzero.

Uφ “!

m P X | φ “a

bwith bpmq ‰ 0

)

.

Definition 104. The sheaf of regular functions on X “ maxSpecpAq is thesheaf OX defined by

OXpUq “ tφ P CpXq | U Ď Uφu .

Fact 105. OXpXq “ A.

A consequence of this fact is also that OXpUf q “ Arf´1s.We now have all the information we need to define the category of affine

varieties! For any two C-algebras A,B and a homomorphism h : A Ñ B wehave a continuous map f : Y “ maxSpecpBq Ñ X “ maxSpecpAq that comesfrom h. This also gives a map

f˚ : OXpUq Ñ OY pf´1pUqq

for each U Ă X “ maxSpecpAq open. These data give a morphism of affinevarieties.

So this category is indeed quite familiar! It’s the opposite of the category offinitely-generated C-algebras.

35


Yesterday we talked about affine varieties. I want to talk about projective va-rieties today, without getting too bogged down in the details. It’s going tobe hard to do, but ideally I will give you an idea of what projective algebraicgeometry while flying overhead.

Euclidean GeometryLocal Global

Open BallsgluingÝÝÝÑ topological space

zero of systems of equationsspaces of orbits for G

œ

M

In Algebraic Geometry, we do something a bit strange. We take the zero setsof systems of equations, which is a global object in Euclidean space, and makethese our local notions. We have no implicit function theorem here, so assign-ing global coordinates is more difficult. So instead larger objects are generallymade by gluing local patches.

Algebraic GeometryLocal Global

Affine Variety Projective Varieties

SpecpAqgluing viaÝÝÝÝÝÝÑregular fns

ProjpAq

Zariski Topology Spaces of OrbitsOX OX

As usual, we won’t talk about Cˆ acting on affine varieties but instead Cˆacting on the functions on an affine variety. We assume that every algebra is adomain.

Definition 106. A Cˆ-linearlized C-algebra A is an action of Cˆ on A (writtenλpxq for λ P Cˆ) such that

(a) the Cˆ-invariants tx P A | λpxq “ xu in A are just C;

(b) the standard character space A1 “ tx P A | λpxq “ λxu (the space onwhich Cˆ acts) generates A as a C-algebra, and in addition A1 is finitedimensional.

Since every representation of Cˆ is multiplication by λk for some k, thenwe break up A as the spaces invariant under each representation as

A “ C‘A1 ‘A2 ‘ . . .

where Ai “ tx P A | λpxq “ λkxu.

36

Example 107. Let A “ Crx1, . . . , xns, and note that A decomposes as

Crx1, . . . , xns “8à

m“0

Crx1, . . . , xnsdeg“m

Then the action of Cˆ on the generators λpxiq “ λxi is ordinary multiplication.The space A1 “ Crx1, . . . , xnsdeg“1 is the space corresponding to k “ 1, andgenerates A (because it includes all xi).

Definition 108. The irrelevant ideal of a graded ring

A “8à

m“0

Ai

is the ideal

m “8à

mą0

Ai.

This is the maximal Cˆ invariant ideal.

Definition 109.

maxProjpAq “ tmaximal Cˆ-invariant homogeneous idealsmx Ĺ mu

Similarly, we can put a Zariski topology on maxProjpAq by declaring theclosed sets to be

ZpIq “ tmX | I Ă mxu

for Cˆ invariant ideals I .

Example 110. If A “ Crx1, . . . , xns, we know that SpecpAq is geometricallyCn. The points in the space maxProjpAq correspond to the lines through theorigin in Cn, that is, mx “ Ip`q for some line ` through the origin. For a point~a “ pa1, . . . , anq, the ideal correpsonding to the line through the origin and ~a is

mr~as “ xxiaj ´ aixj | i, j “ 1, . . . , ny.

The irrelevant ideal corresponds to the origin, which doesn’t define a linein projective space. We throw it out much like we throw out the origin whendefining projective space.

So now we need to know the functions on this space. We define the rationalfunctions on maxProjpAq by ratios of homogeneous polynomials of the samedegree.

CpAq0 “"

F

G

ˇ

ˇ

ˇ

ˇ

F,G P Am for some m*

.

These indeed define functions on the space, but I leave that as an exercise.

37

Problem 111. Show that φpmr~psq “ φpmod mr~psq makes sense as a complexnumber.

As before, let Uφ be the domain of φ. Define a sheaf on X “ maxProjpAq by

OXpUq “ tφ P CpAq0 | U Ď Uφu.

We can also get at these spaces by gluing affine varieties together. How dothe affine spaces sit inside this projective space? Let a P A1. Then the domainwe want for the corresponding variety is

Ara´1s0 “

"

F

ad

ˇ

ˇ

ˇ

ˇ

F P Ad

*

Ă CpAq0.

Hartshorne calls this Apaq. Note that CpAq0 is the same as the field of fractionsof CpAra´1s0q.

The affine variety corresponding to this algebra is maxSpecpAra´1s0q, whichis in bijection with the open set maxProjpAqzZpaq. The bijection is given by

xai{a´ piy ÐÑ xa´ aipiy,

where ai generate A1 as a C-vector space.Given a Cˆ-linearized C-algebra with A1 generated by a1, . . . , an as a C-

vector space, we have a surjective map CrA1s�A. We can think of CrA1s aslines through the origin in the dual vector-space of A1, which we call P1pA1q.This contains maxProjpAq.

So what have we done? Given a Cˆ-lienarized C-algebra A, we have pro-duced a projective variety. This comes with a sheaf of functions on it that islocally affine. In affine space, it’s easy to go back from the variety to the alge-bra A, but in projective space, this is not the case. Given a projective variety,there are infinitely many ways to make it from a Cˆ-linearized C-algebra. Soit’s a difficult question to determine when projective varieties are isomorphic.

Example 112. Let d be a fixed integer, and let A be a Cˆ-linearized C-algebra,

A “ C‘A1 ‘A2 ‘ . . .‘Ad ‘ . . .

What happens when we take only multiples of d?

C‘Ad ‘A2d ‘ . . .

This carries an action of Cˆ. What’s maxProj of this space? It’s just the Veroneseembedding. We have to rescale by setting µ “ λd.

If A and B are both Cˆ-linearized C-algebras, what space does A bC B

determine? It’s the product of maxProjpAq and maxProjpBq.

C‘A1 bB1 ‘A2 bB2 ‘ . . .

38

We can get the Grassmannian Grpm,nq in this way too. If A “ Crtxijus whereX “ txiju are the Grassmann coordinates, then we can think about the ring

C‘ xdetpXI1qy ‘ xdetpXI1qy ‘ . . .

where Ii are all size m subsets of t1, 2, . . . , nu.


Today I want to talk about flag manifolds and (hopefully) introduce quivervarieties so you can see what the big kids in the other room (graduate studentseminar) are talking about.

Let’s talk about the characteristic polynomial. Tom already talked aboutthese things, but I’m going to talk about them again. This feeds right into therepresentation theory and Borel-Weil.

The characteristic polynomial is an invariant of SLpn,Cq acting on n ˆ n

matrices A “ paijq by conjugation.

detpxI ´Aq “ xn ´ trpAqxn´1 ` . . .` p´1qn detpAq,

and the intermediate terms come from sums of principal minors of the matrix.If you’ve never worked it out before, you should do so. We can also write thisas

detpxI ´Aq “ xn ´ p1pAqxn´1 ` . . .` p´1qnpnpAq,

for some polynomials pi in the coordinates of A. This is the ring of invariantsof this action.

Cry1, . . . , yns “ CrtaijusSLpn,Cq

yk ÞÑ pkpaijq

Using our algebraic geometry technology, we get a map on the spectra ofthe rings

q : maxSpecpCrtaijusq Ñ maxSpecpCry1, . . . , ynsqxxij ´ aijy ÞÑ xyk ´ pkpaijy

But we can think of this as sending a point (maximal ideal) associated to thematrix A to a point in Cn given by

A ÞÑ ptrpAq “ p1pAq, . . . , pnpAq “ detpAqq

Obeserve that q is onto: given a vector in Cn we can always cook up amatrix in SLpn,Cqwhose coefficients of the characteristic polynomial are givenby these coordinates.

39

Definition 113. The set of matrices which are upper triangular with 1’s on thediagonal is

N “

"

upper triangular matrices with 1 on diagonals„

1 ˚

0 1

*

Each of these is has the identity matrix in the closure of its orbit underconjugation by SLpn,Cq.

Example 114.„

t 0

0 t´1

„

1 a

0 a

„

t´1 0

0 t

“

„

1 at2

0 1

tÑ0ÝÝÝÑ

„

1 0

0 1

Let’s try to tie all of this back to representation theory. Given a finite dimen-sional representationn of SLpn,Cq,

ρ : SLpn,Cq Ñ GLpV q,

the character only depends on the conjugacy class of an element of g P SLpn,Cq.If gt P N , then

trpρpgtqq “ trpρpidqq.

Since characters distinguish representations, and we can’t tell the differencebetween conjugation by N and conjugation by the identity, then all of the irre-ducible representations of SLpn,Cq lie inside the invariant ring of the regularrepresentation.

CrSLpn,CqsN

Now let’s consider SLpn,Cq

œ

CrSLpn,CqsN . This is related closely to theflag varieties we talked about earlier. Recall that the flag variety Flpnq is the setof flags in Cn.

Flpnq “ tV1 Ă V2 Ă . . . Ă Vn´1 Ă Cnu

Now fix the standard flag

xe1y Ă xe1, e2y Ă . . . Ă xe1, . . . , eny.

SLpn,Cq acts on Flpnq by multiplying the basis vectors. What is the stabilizerof the standard flag? It has to fix subspaces xe1, . . . , eky, so the matrix needs tobe upper triangular.

Definition 115. The Borel subgroup of a Lie group is the stabilizer of a flag. Inour case, this is upper triangular matrices

B “

$

’

’

’

&

’

’

’

%

»

—

—

—

–

˚ ˚ . . . ˚

0 ˚ . . . ˚

.... . .

...0 0 . . . ˚

fi

ffi

ffi

ffi

fl

,

/

/

/

.

/

/

/

-

40

So we can think of the flag variety as Flpnq – SLpn,Cq{B. Now let

T “ B{N “ tpt1, . . . , tnq | t1t2 ¨ ¨ ¨ tn “ 1u.

This is often called the torus of this Lie group.We want to understand A “ CrSLpn,CqsN . This is a C-algebra, as last lec-

ture, and we have an action of T on A.

Example 116. For n “ 2, T “ Cˆ, and

A “ C‘A1 ‘A2 ‘ . . .

We also have thatmaxProjpAq “ Flp2q “ P1C.

This makes sense, because these things are just flags in C2 (so lines through theorigin)! The Borel-Weil theorem says that this is all of the representations ofSLp2,Cq.

Example 117. In general, we have an action of T “ pCˆqn´1 on CrSLpn,CqsN .

Remark 118.

maxSpec`

CrSLpn,CqsN˘

–SLpn,Cq{N

TÝÝÝÝÑ Flpnq

This whole discussion is a machine for generating representations of SLpn,Cq.The procedure is as follows

(1) take a character χ of the torus T ;

(2) we get a C-algebra CrSLpn,CqsNχ with an action of Cˆ;

(3) take maxProj of this space to get the representation on the flag varietyFlpnq.

In our last ten minutes, let’s talk briefly about quiver varieties.

Definition 119. A quiver is a directed graph.

Example 120. Here’s a quiver that Aaron Bertram studied for quite some time.It’s the quiver for the projective plane.

‚ ‚ ‚

Definition 121. A quiver representation is an assignment to each vertex of thequiver a vector space and to each arrow a directed graph. In addition, thesearrows should commute.

41

Example 122. To the quiver from example 120, we might have a representation

V1 V2 V3

with some maps T1,2,3 : V1 Ñ V2 and S1,2,3 : V2 Ñ V3.

Let Q be a quiver with representation V . Choosing a basis for each of thesevector spaces in a quiver representation, the maps become matrices. We canact on these spaces by change of basis via

GLpV1q ˆGLpV2q ˆ ¨ ¨ ¨ ˆGLpVnq{Cˆ

The quiver variety is the spectrum of the invariant ring of the quiver algebra,that is, the quiver variety is

Spec´

CrQsSLpn1qˆ¨¨¨ˆSLpnkq¯

These are just some things that are big in algebraic geometry right now.Flag varieties, quiver varieties, Grassmannians, etc.

42

Flag Varieties and...

Documents

Transcript of Flag Varieties and...