Flag Varieties and Representations

Home , Flag (linear algebra)

Lectures by Aaron Bertram PCMI 2015 Undergraduate Summer School

Notes by David Mehrle [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 Lecture 1 June 30, 2015

Yesterday Tom Garrity said that “functions define the world” (and we all learn it as children). The better version is that “categories define our universe”). Action of a group is always inside a category. We will have a categorical moment 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:

f g g˝f 1. Morphisms compose correctly. If A ÝÑ B and B ÝÑ C then A ÝÝÑ C is another morphism; 2. Composition is associative;

f 3. For each object X there is an identity arrow idX such that for any X ÝÑ Y , g f ˝ idX “ f and for any Z ÝÑ X, idX ˝ g “ g.

f Deﬁnition 2. In any category, A ÝÑ B is an isomorphism if f has an inverse, g i.e. there is a morphism B ÝÑ 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/ right inverses, 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 is an isomorphism. An example of a group is AutpXq “ tautomorphisms of Xu for any object X of any category. We will talk about several categories, starting with the category of sets, and then 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’ll call 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 commute, and automorphisms are just permutations. The symmetric group is Sn “ Autprnsq. (Which is Permpnq in Aaron’s notes but that’s stupid).

2 Deﬁnition 4. Here’s the cycle notation for permutations. For a permutation 1 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 is p1qp2qp3qp4qp5q. 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 fpjq ´ fpiq sgnpfq :“ j ´ i iăj ź 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, 1 ´ 2 3 ´ 2 3 ´ 1 sgnpp1 2qq “ ¨ ¨ “ ´1. 2 ´ 1 3 ´ 1 3 ´ 2 3 ´ 2 1 ´ 2 1 ´ 3 sgnpp1 2 3qq “ ¨ ¨ “ 1 2 ´ 1 3 ´ 1 3 ´ 2 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 denominator 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 in the denominator at some point cancelling the fpiq ´ fpjq. This happens for each i, j. |fpjq ´ fpiq| iăj “ 1. j i ś iăj | ´ | ś 3 3. σpτpiqq ´ σpτpjqq sgnpσ ˝ τq “ j ´ i iăj ź σpτpjqq ´ σpτpiqq τpjq ´ τpiq “ τpjq ´ τpiq j ´ i iăj ź σpτpjqq ´ σpτpiqq τpjq ´ τpiq “ “ sgnpσqsgnpτq τpjq ´ τpiq j ´ i τ i τ j iăj p qăźp q ź 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, sgnpσq “ 0 σP§ ÿn Lecture 2 July 1, 2015

I’m going to start with a great line. Almost all algebraic geometry talks begin with 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 respect to the p-adic norm | ¨ |p for p a prime (p is always prime, except on wikipedia where p “ 10). Q Ă R n 1 Here’s how the p-adic norm works. For n an integer, |p |p “ pn , |1|p “ 1, a a and if b P Q such that p - a and p - b, then | b |p “ 0. Completing with respect to this norm gives the p-adic numbers Qp, also a field. And it too has an algebraic closure. Q Ă Qp Of course, we also have the algebraic closure of R, which is the familiar complex numbers C. We can also take the algebraic closure of Q to get the algebraic numbers Q. Fields between Q and Q are number fields k, if they are finite extensions of Q.

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

There are finite fields too, Fp “ Z{pZ, and Fpn Ą Fp. We can also take algebraic 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 functions pptq Cptq “ p, q P Crts , qptq " ˇ * ˇ or it’s algebraic closure. Or the field of Laurentˇ series ˇ 8 i Cpptqq “ ait ai P C . # ˇ + i“ń ˇ ÿ ˇ ˇ It’s algebraic closure is the Puiseaux series Cˇ pptqq. Maybe you have two variables, with a field Cpt1, t2q. Or n variables, such as Cpt1, . . . , tnq. In some sense all of algebraic number theory lives inside the algebraic closure of one of these things. 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 that k “ C or k “ R. Last time we talked about the category of sets, Sets. Let’s define the product of two sets categorically. Definition 10. The product of two sets S, T is a set U together with maps p: U Ñ S and q : U Ñ T such that for any other set Z with maps f : Z Ñ S and g : Z Ñ T , there is a unique map u: Z Ñ U such that p ˝ u “ f and q ˝ u “ g.

Z u f g U

p q S T Now let’s talk about vector spaces over a ﬁxed ﬁeld k. We have a category called k-Vect of vector spaces over k, where the objects are k-vector spaces and the morphisms are linear maps. k--Vect is an abelian category, which in this case tells us that

HompV,W q :“ tlinear maps f : V Ñ W 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 n ˆ m 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 a group (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 :“ sgnpσq aiσpiq. σPS i“1 ÿn ź Next, some representation theory. We’ll commit the cardinal sin of choosing n a basis e1, . . . , en for k .

n Deﬁnition 12. The permutation representation of Sn on k corresponds to the action of Sn on vectors by permuting the coordinates. This is given by a map n Sn Ñ GLpk q, σ ÞÑ Pσ, where Pσ is the permutation matrix with columns given by Pσei “ eσpiq.

n n Problem 13. Given a vector ~v “ i“1 viei P k , then what is Pσ~v ?

Note that the determinant of ař permutation matrix Pσ is the sign of its permutation: detpPσq “ sgnpσq. Deﬁnition 14. A matrix A is semisimple if and only if there is an invertible matrix L such that LAL´1 is diagonal. This is exactly when A has a basis of eigenvectors. 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

λ ´ cos θ sin θ det “ pλ ´ cos θq2 ` sin2 θ ´ sin θ λ ´ cos θ „  “ λ2 ´ 2 cos θλ ` cos2 θ ` sin2 θ ? 2 cos θ ˘ 4 ´ 4 cos2 θ ùñ λ “ “ cos θ ˘ i sin θ “ eiθ 2 Let’s ﬁnd the eigenvectors too. We want the kernel of

eiθ ´ cos θ sin θ i sin θ sin θ “ , ´ sin θ eiθ ´ cos θ ´ sin θ i sin θ „  „ 

6 which is spanned by 1 1 and , ´i i „  „  corresponding to eiθ and e´iθ, respectively.

Lecture 3 July 2, 2015

First, a few comments about yesterday’s lecture. At the very end, we said that in C2, p1, iq K p1, íq. This is not true for the ordinary dot product. We need to use 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 can form 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 problems we’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 group actions. Some examples of groups we’re going to talk about include AutpXq for X 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 category in which the objects are groups and the morphisms are group homomorphisms. Example 18. The exponential map is a group homomorphism from pR, `q to x`y x y pRą0, ¨q, because e “ e e . It has an inverse log, but if we instead had the exponential 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 interesting commuting diagram of groups.

φ Sn GLnk

sgn det t˘1u k˚

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

x1, . . . , xn w1px1, . . . , xnq “ w2px1, . . . , xnq “ ... “ wmpx1, . . . , xnq “ 1 . B ˇ F ˇ ˇ ˇ 7 Example 21. The free group on two generators is given by the finite presentation 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 normal subgroup that creates the relations. n Example 22. The cyclic group Cn :“ xx | x “ 1y. The integers, Z – xxy is the infinite cyclic group. n 2 ´1 Example 23. The dihedral group Dn :“ xx, y | x “ y “ xyxy “ 1y. It’s not immediately 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. Here is 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 2 3 according to the relations xi “ 1, xixj “ xjxi if |i ´ j| ě 2, pxixi`1q “ 1. We can write this group as a Dynkin diagram by putting one dot per variable and 2 a line between two dots if the corresponding variables satisfy pxixjq “ 1. ‚ ‚ ‚ ‚ ‚

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

Problem 25. The cyclic group Cmn is isomorphic to Cm ˆ Cn if and only if gcdpm, nq “ 1. Here’s a really cool theorem about finitely generated abelian groups. Theorem 26. Let A be a finitely generated abelian group. Then r A – Z ˆ Z{d1Z ˆ Z{d2Z ˆ ¨ ¨ ¨ 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 subgroup B of an abelian group A, we have a quotient group A{B (for arbitrary groups, it only makes sense for normal subgroups). Okay let’s finally talk about group actions. An action of a group G on an object X in some category C can be defined in several ways, but the following is the cleanest. Definition 27. A group action of a group G on X P C, where C is a category, œ is a homomorphism φ: G Ñ AutpXq. We often write this G X. If X “ V is a vector 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 such that 1 ¨ x “ x for all x P X and g1pg2xq “ pg1g2qx.

8 Lecture 4 July 3, 2015

Today we’re going to talk a little bit more about categories before we move on to G-modules. Deﬁnition 29. Given two categories C and D, a (covariant) functor is a map F : C Ñ D that assigns to each object c P C an object F pcq P D, and to each arrow f : c Ñ c1 in C, an arrow F pfq: F pcq Ñ F pc1q in D, subject to the following conditions:

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 functor that forgets the structure of some object and regards it as a more basic one. For example, you can forget that a vector space is a vector space and think of it as a 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 need not 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 representation theory. Example 31. Let’s find the character table for Z{3Z. We first have the trivial representation, and then we have the rotations by 2π{3 as another representation. What’s the third one? It’s an abelian group, so has three conjugacy classes and therefore three irreducible representations. Try as hard as we can, we can’t find the final representation over the real numbers. We have to move to the complex numbers. Let ω “ e2πi{3. The character table for Z{3Z is below.

Z{3Z t0u t1u t2u trivial 1 1 1 2 χ1 1 ω ω 2 χ2 1 ω ω

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

n x~x,~yy :“ xiyi. i“1 ÿ 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’s pick a group action φ: G Ñ 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 by φpgqphq “ gh. This is an example of a transitive action. Note that we think of AutpGq as the automorphisms of G as a set, not as a group. Notice that φpgqph1h2q “ gh1h2 but φpgqph1qφpgqph2q “ gh1gh2, so φpgq is not an automorphism for each g.

Example 34. The conjugation action is ψ : G Ñ AutGroupspGq given by ψpgqphq “ ghg´1. This differs from the previous example because here, ψpgq is an automorphism 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 of G, by a new action ψ1 : G Ñ AutptH | H ď Guq deﬁned as ψ1pgqpHq “ gHg´1.

Deﬁnition 35. H ď G is normal if H is stabilized by G under the conjugation action. We write H C G 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 generated by p1 2qp3 4q, p1 3qp2 4q, p1 4qp2 3q, and it is normal. This is unique to the symmetric group S4. For any other symmetric group Sn for n ‰ 4, the only normal 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: G Ñ G is a homomorphism then ker φ ď G is normal.

10 Now we are equipped to talk about the lifting problem. Given V a vector space and U a subspace, we want to ﬁnd a subspace W of V such that W is isomorphic to V {U.

V V {U

Lecture 5 July 6, 2015

Let G be a group.

Deﬁnition 38. A G-module is a vector space V together with a homomorphism φ: G Ñ GLpV q.

That is, a G-module is a representation of G. To write things concisely, we œ sometimes write G V . Additionally, we write the action of G as g ¨ ~v rather that φpgqp~vq.

Deﬁnition 39. In the category of G-modules, which we call G-Mod, the objects œ are G-modules G V and the morphisms are the intertwiners f : V Ñ W such that fpg~vq “ gfp~vq. We call this property G-linearity.

A cute remark (taking what Tom did and putting it into weird notation to obscure it) is that there is a forgetful functor F : G-Mod Ñ C-Vect which forgets 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 between 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 ﬂ W ;

(b) HomG-ModpV,W q – C if V – W . œ œ 3 Example 41. S3 C by permuting the basis vectors, and S3 C by the trivial representation. 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 kernel of f. 3 ker f “ t~v P C | v1 ` v2 ` v3 “ 0u.

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

3 f S3 C C S3

ker f

What’s the matrix for the permutation p1 2q? Check it’s action on the basis vectors. p1 2q ¨ pe1 ´ e2q “ e2 ´ e1 “ ´pe1 ´ e2q

p1 2q ¨ pe2 ´ e3q “ e1 ´ e3 “ pe1 ´ e2q ` pe2 ´ e3q So the matrix is given by ´1 1 p1 2q “ 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 that this representation is isomorphic to the other realization of the 2-dimensional irreducible representation of S3 we found earlier.

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

1 z z ÞÑ . 0 1 „  This is a very important example! Similar to the Heisenberg group. The only stable line is the line y “ x.

Theorem 44 (Maschke’s Theorem). Every representation of a ﬁnite group over R or C is a direct sum of irreducible subrepresentations. Each representation decomposes into irreducibles. A more sophisticated version says that kG is semisimple so long as the characteristic 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 solve the lifting problem of finding a subspace of V that is isomorphic to V {U. This will be an orthogonal compliment, but not with respect to the usual Hermitian inner product on Cn. Let’s first talk about what inner product we’re going to use here. The Her- mitian inner product is n ~v ¨ ~w “ viwi. i“1 ÿ

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 1 x~v, ~wy “ pg~vq ¨ pg ~wq. |G| gPG ÿ Now we have a genuine G-invariant bilinear form on V . Then we take W “ U K, 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 P W, ~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 since U is G-invariant, so now we know that h~w P W , 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 , pull out an irreducible component U, and then write V as V “ U ‘ U K. Now do it again with U K, 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 representations of a group? We get something called the regular representation.

Deﬁnition 45. The regular representation of a group G is the vector space with basis xeg | g P Gy, with an action of G given by h ¨ eg “ ehg. This is also the group-algebra CrGs, and it has dimension equal to the order of G, dim CrGs “ œ œ |G|. We can extend the action G CrGs linearly to an action CrGs CrGs. Here’s a cool construction that shows that each irreducible representation appears at least once in CrGs. Given a decomposition into irreducibles,

CrGs “ U1 ‘ U2 ‘ ... ‘ Un, œ and an irreducible representation G U. , choose ~u P U. Construct the function 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 bunch of maps on each irreducible component fi : Ui Ñ U. By Schur’s Lemma, each one 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 some i. What about the converse? Now we want to know how many times each irreducible representation appears. In particular, how often is U – Ui for each i? The answer is that a irreducible representation U appears dim U-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 an isomorphism given by f ÞÑ fpeidq and its inverse ~u ÞÑ f~u. Then,

U – HompCrGs,Uq n – Hom Ui ,U ˜i“1 ¸ n à – HompUi,Uq i“1 à Now by Schur’s lemma, HompUi,Uq – C if Ui – U, or zero otherwise. And these are isomorphic, so there must be dim U-many nonzero items on the right. This comes from a rather silly observation, that

n dim U U – HompUi,Uq – C. i“1 i“1 à à Lecture 6 July 7, 2015

Today, our groups are no longer finite (unless you maybe like to work over finite fields). They’re going to be groups of matrices, LieGroups. Today we’re going to talk about examples, and then for the next few days we’ll talk about what it really means to be a Lie group. We’ll talk about differential geometry and eventually, algebraic geometry, because the Lie groups we care about are algebraic. For now, all I have to do is convince you that Lie groups are interesting. For now, let’s let this ad-hoc definition suffice: Definition 46 (Temporary). A Lie group is a group of matrices defined by a polynomial condition on GLnk, where k is either R or C.

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

GLnC “ glnCztX P glnC | det X “ 0u

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

n paijq , pbijq ÞÑ aikbkj ˜k“1 ¸ ÿ And there is an inversion map i: GLnC Ñ GLnC given by Kramer’s rule. Both m and i are polynomial maps in the entries of matrices A and B.

14 Example 48. a b GL “ ad ´ bc “ 0 2R c d "„  ˇ * ˇ If you think about this really hard, you willˇ see that this is like a cone. Because if ad ´ bc “ 0, then so does λpad ´ bcq “ 0 ˇas well for any λ, and it’s singular at the 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 orthogonal matrices. n On “ tA P GLnR | A~v ¨ A~w “ ~v ¨ ~w for all ~v, ~w P R u We can reinterpret the length-preserving condition as AT A “ I, because ~vT ~w “ ~v ¨ ~w “ A~v ¨ A~w “ pA~vqT A~w “ ~vT pAT Aq~w

T Choosing ~v “ ~ei and ~w “ ~ej, we see that pA Aqij “ δij for all i. The rows and columns of orthogonal matrices are of unit length and orthogonal. Example 51. The length-preserving transformations of Cn are the unitary matrices. n Un “ tA P GLnC | xA~v, A~wy “ x~v, ~wy for all ~v, ~w P C u. By the same argument as in the previous example, we see that unitary matrices must satisfy A˚A “ I. The rows and columns of unitary matrices are of unit length and orthogonal with 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 with only 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 unitary matrix to some value on the unit circle. In dimension 2, we have

cos θ ´ sin θ SOp2, q “ R sin θ cos θ "„ * a ´b z ` z´1{2 ´pz ´ z´1{2iq SOp2, q “ a2 ` b2 “ 1, a, b P “ – ˚ C b a C ´pz ´ z´1{2iq z ` z´1{2 C "„  ˇ * "„ * ˇ ˇ We should talk aboutˇ SUp2q at the same time we talk about SOp3, Rq. As the rotations 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. This is because the characteristic polynomial looks like

x3 ´ trpAqx2 ` bx ´ detpAq, and detpAq “ 1. This polynomial has three roots, and one of them must be real because all of the coefﬁcients are real. These roots multiply to 1, so the real root must be ˘1. Once we know that there is an eigenvector with eigenvalue 1, this corresponds 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 ć ` di 1 0 i 0 0 ´1 0 1 “ a ` b ` c ` d c ` di a ´ bi 0 1 0 í 1 0 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!

2 2 2 2 H “ ta ` b~i ` c~j ` d~k | a ` b ` c ` d “ 1u

This is the equation for a sphere in R4, so SUp2q is topologically the 3-sphere S3. Another fun fact. On the sphere S3 there is an action of SUp2q for which the latitudes of SUp2q are conjugacy classes and the longitudes are stabilizers! SUp2q is a double cover for SOp3, Rq as a manifold.

16 Lecture 7 July 8, 2015

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 that 2 they’re sitting inside the Euclidean space Cn and are described by polynomial equations on the coordinates. For instance, SLpn, Cq is described by the equation detpAq “ 1. Not Compact Compact (Uncool) (Cool) SLpn, Cq SUpnq SOpn, Cq SOpn, Rq From here we’re going to try and understand these things as manifolds. Today we’re going to do some topology. Friday will be function orgy day.

Deﬁnition 53. A topology on a set X 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 implies iPI Ui is open.

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

U “ Bri ppiq. iPI ď Remark 55. I’m going to say “balls” a lot. I’ll try not to giggle too much. Sorry about that.

Example 56. Let X be the real line with two origins, pictured below. This can be described as R Y 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 between them.

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

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

17 Example 59. Polynomials P px1, . . . , xnq P Rrx1, . . . , xns are continuous functions P : Rn Ñ R with respect to the Euclidean topology. If Y is a subset of X, and X is a topological space, we can deﬁne an induced topology on Y by stating that the open sets of Y are exactly pU X Y q for U open in X. 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 is continuous if and only if i ˝ f : Z Ñ X is continuous.

X i˝f i Z Y f

This fact and the previous example allows us to conclude that all of our ex- 2 amples of Lie groups are topological spaces: they are subspaces of Cn defined by 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 that x1 P U1, x2 P U2 yet U1 X U2 “H;

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

• compact if for all open covers X “ αPA Uα, there is a ﬁnite subcover n X “ Uα ; i“1 i Ť • a manifoldŤ 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 Euclidean topology is compact if and only if X is closed and bounded.

2 Example 62. Let’s think about SOpn, Rq as a subset of Rn . Each matrix in SOpn, Rq is of the form

| | | A “ ~u1 ~u2 . . . ~un , » ﬁ | | | – ﬂ 2 which is a vector in Rn . Each of the columns is orthogonal to each other, so it 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 furthest ? distance we travel from the origin is n. Additionally, SOpn, Rq is closed because it is the inverse image of a closed set 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 contains an unbounded line (well, excluding the origin) t 0 t ÞÑ . 0 1 „ t  n n Here’s an example of gluing. Let U1 “ R and let U2 “ R . Let U12 “ n n n n R zt0u and let U21 “ R zt0u. Glue by the identity id “ f12 : R zt0u Ñ R zt0u to get an n-space with two origins. This is not Hausdorff. Every open set containing one origin intersects every open set containing the other. If instead we glue along the map ~v f12p~vq “ , }~v}2 then we get the sphere. One origin becomes a point at 8 and we get something homeomorphic to Sn in Rn`1. This is the stereographic projection.

Lecture 8 July 9, 2015

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

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 it Gp2, 4q. The elements of Gp2, 4q are H

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

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

X X X ù X ù X X ù X X X

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

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

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

n n P C “ tpx0 : x1 : ... : xnq P P C | xi ‰ 0 for some iu, where px0 : x1 : ... : xnq means we consider points up to (nonzero) scalar multiplication.

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

px0{x1, 1q x1 ‰ 0 px0 : x1q “ #p1, x1{x0q x0 ‰ 0

We can make the two cases into two copies of the complex numbers. Call them 1 U0 and U1, where U0 is the subset of P C where x0 ‰ 0 and U1 is the subset where 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, 1 we identify a point in U1 with its image under f. This gives us CP as two overlapping copies of C, each missing a point that the other contains. This last point 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 projective space. In P2C, thinking about how to get all the lines through the origin in this way 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 the line, and finally a 3-space. We get this from first setting z “ 1, and from this we get a bunch of lines but miss all those in the xy-plane because they don’t pass through the plane z “ 1 3 in C . This corresponds to the subspace xe0, e1, e2y; this is lines in xe0, e1, e2y yet not in xe0, e1y. Then by setting y “ 1, we add all of the lines through the origin except 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 of lines in xe0y. The generic behavior is

Lines Subspace Coordinates 2 ` R xe0, e1y C 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 in dimension 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 dimension m. n Gpm, nq “ tW Ď C | dim W “ mu. Much like projective space, we want to give coordinates to points in the Grassmanian. We will do this by describing the points with m coordinates. If n the coordinates of C are x1, . . . , xn, then a point is described by

pxi1 , . . . , xim q for an index set I “ p1 ď i1 ă i2 ă ... ă im ď nq, each 1 ď ij ď n. In projective space we set one coordinate to 1, but in a Grassmanian we set m coordinates to 1. There is a much more sane way to describe these coordinates, much like the n ratios px0 : ... : xnq in P C. Definition 67. The Grassmann coordinates for a point in Gpm, nq is an m ˆ n matrix x11 x12 . . . x1n . . . . X “ » . . .. . fi x x . . . x — m1 m2 mn ffi where the basis of a space–W is the rows of this matrix.fl We consider such matrices as equivalent up left-multiplication by GLpm, Cq, which only changes the basis but doesn’t leave the space W . Definition 68. From the Grassmann coordinates, we can define the Pl ücker n ´1 Coordinates as points in P pmq by taking determinants of minors

detpMI1 q: detpMI2 q: ... : detpMI n q , pmq ˆ ˙ where Ii Ă t1, . . . , nu are all the distinct subsets of t1, . . . , nu and MI is the minor of the Grassmann coordinate matrix X indexed by I. p n q´1 This also gives us the Pl ¨uckerembedding into P m . Example 69. For Gp2, 4q, the Plucker¨ x x 1 0 11 12 ÝÝÝÝÑpPlucker¨ x x ´ x x : ´ x : x : x : 1q x x 0 1 11 22 12 21 21 11 12 „ 21 22  Because all of the Grassmann coordinates appear, this map is injective!

Lecture 9 July 10, 2015

The Grassmannian – Done Right Recall that n Grpm, nq “ tW Ă C | dimpW q “ nu.

22 We also have a Plucker¨ embedding

p n q´1 Grpm, nq ãÑ P m C.

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

X X X X X X m rows X X "

n´m columns

n looooomooooon Problem 71. There are m 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

c λ “ 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 if and only if µ “ λ ` some squares . To begin with these Schubert cells, we ﬁx a full ﬂag

n 0 Ă xe1y Ă xe1, e2y Ă ... Ă xe1, . . . , en´1y Ă C

For ease of notation, call it

n 0 “ V0 Ă V1 Ă V2 Ă ... Ă Vn´1 Ă Vn “ C

What is the generic behavior of a W Ă Cn of dimension m with respect to this ﬂag? We might have dimpW XV1q “ 0,..., dimpW XVn´mq “ 0, dimpW XVn´m`1q “ 1,... dimpW XVnq “ 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,..., 0, 1, 2, 3, . . . , mq.

n´m We can also look at the jumploooomoooon sequence of successive differences in dimension. For the above, this is p0, 0,..., 0, 1, 1, 1,..., 1q.

n´m By some linear algebra shenanigans,loooomoooon observe that the vectors of V which lie in W X Vn´m`1 look like p˚, ˚,..., ˚, 1, 0, 0 ..., 0q

n´m with respect to the basis thatloooomoooon comes from our ﬂag. And likewise, we can force the vectors of V which lie in W X Vn´m`2 to look like p˚, ˚,..., ˚, 0, 1, 0 ..., 0q.

n´m Putting these vectors intoloooomoooon a matrix, we get the Grassmann coordinates! A point in W looks like

˚ ... ˚ 1 0 ... 0 ˚ ... ˚ 0 1 ... 0 »...... fi ...... — ffi —˚ ... ˚ 0 0 ... 1ffi — ffi – fl Now what about a non-generic subspace W ? This means that the sequence of 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 – the dimension of intersections increases by 1 a total of m times. Example 72. What about Grp2, 4q? We calculate with respect to the standard ﬂag 4 0 Ă xe1y Ă xe1, e2y Ă xe1, e2, e3y Ă C . 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 the matrix ˚ 1 0 0 . ˚ 0 ˚ 1 „ 

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

1001

0011 0101 1010 1100

0110

Look familiar? The jump sequences correspond to Young diagrams as follows. The most generic sequence corresponds to the empty Young diagram. Then for any other jump sequence, observe how many positions the left 1 had to 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 the next row, and so on. The biggest Young diagrams correspond to the smallest subspaces in the stratiﬁcation of the Grassmannian. So for example,

Example 73. The jump sequence p0, 0, 1, 1q corresponds to an empty Young diagram. 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 n F s subspaces W Ă C corresponding S “ λ to λ with respect to the flag F " *

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

Sλ “ Sµ λďµ ď What happens if we reverse the ﬂag? If we take the ﬂag

n 0 Ă xe1y Ă xe1, e2y Ă ... Ă xe1, . . . , en´1y Ă C and substitute the ﬂag

n 0 Ă xeny Ă xen, en´1y Ă ... Ă xen, . . . , e1y Ă C , 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 1 ˚ ˚ 0 ù 0 ˚ ˚ 1 0 0 0 1 „  „  These complimentary Schubert cells intersect in a point! This is Poincare´ duality.

Lecture 10 July 13, 2015

This week, we’ll talk about some algebraic geometry related to the Lie groups that 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

n P C, Grpm, nq, Flpnq, which are both algebraic and compact. The rightmost one above is the flag variety. As Tom Garrity said in his first lecture, “functions describe the world.” This is very true of algebraic geometry, where everything eventually gets replaced with functions rather than objects. For instance, we can realize this slogan in the group algebra CrGs. All representations of a finite group G are contained inside CrGs, which we can think of as functions on G, with multiplication convolution.

CrGs “ tf : G Ñ Cu “ fpgqg . #gPG + ÿ

26 Let M be a manifold. Because M is a topological space, we know what it means for function f : M Ñ R to be continuous. But what does it mean for f to be differentiable? We can deﬁne this in terms of the functions on M. œ Remark 76. From an action G M of a group on a manifold, we get an action of G on the functions f : M Ñ R deﬁned by

g ¨ fpxq “ fpg´1xq.

Hierarchy of Functions

There is a hierarchy of functions f : R Ñ R

(1) continuous functions, for which lim |fpx0 ` hq ´ fpx0q| “ 0; hÑ0 (2) differentiable functions, for which

|fpx ` hq ´ y h ´ fpx q| lim 0 1 0 “ 0, hÑ0 |h|

1 for some y1, which we call f px0q;

(3) smooth (inﬁnitely differentiable) functions, for which f 1pxq, f 2pxq, etc. are differentiable as functions of x.

Smooth functions have Taylor series,

h2 fpx ` hq « y ` y h ` y ` ... 0 0 1 2 2

piq ´1{x2 where yi “ f px0q. Of course, there are functions like fpxq “ e that don’t agree 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 for p, q P Rrxs.

Over the complex numbers, the hierarchy goes like this.

(1) Continuous functions;

(2) holomorphic functions (which are complex-differentiable and smooth and analytic) |fpz ` hq ´ y z ´ y | lim 0 1 0 0 “ 0, hÑ0 |h| 1 where we set f pz0q “ y1 and set y0 “ fpz0q.

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

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

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

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

|fp~x ` ~hq ´ ~y ¨ ~h ´ ~y | lim 0 1 0 “ 0. ~hÑ0 |~h|

The derivative is the gradient

Bf Bf ∇fp~x0q “ ~y1 “ p~x0q,..., p~x0q . Bx Bx ˆ 1 n ˙ We can also talk about inﬁnitely differentiable and analytic functions. What about multivariable functions over C? A function f : Cn Ñ C is holomorphic if and only if it is holomorphic in each variable separately. This fact is Hartog’s theorem. The Implicit function theorem tells us that if we have a vector-valued func- ~ n m ~ tion f : R Ñ R and write f “ pf1, . . . , fmq, we can rewrite some portion of ~ the image of f as the zero set of the functions f1, . . . , fm. We can reinterpret this theorem as follows.

Theorem 79 (Implicit Function Theorem). If f~: Rn Ñ Rm or f~: Cn Ñ Cm, then m Vpf~q “ t~x P C | f~p~xq “ 0u is an analytic manifold near any point ~x0 for which

|fp~x ` ~hq ´ Jacpf~qp~x q~h ´ f~p~x q| lim 0 0 0 “ 0, ~hÑ0 |~h|

~ and the Jacobian Jacpfq has full rank near ~x0.

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

28 Example 81. We can use this to analyze the Lie groups from before. Is SLpn, Cq a 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 equation can be found by expanding the determinant along minors Mij corresponding to removing the i-th row and j-th column.

Bf i`j “ p´1q detpMijq Bxij

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

fpI ` εBq “ fpIq ` JacpfqεB ` Opε2q but on the other hand,

fpI ` εBq “ pI ` εBqT pI ` εBq ´ I “ 0 ` εpBT ` Bq ` ε2BT B ` Opε3q, 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 compute 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 is fpAq “ detpAq ´ 1, and near the identity

fpI ` εBq « detpI ` εBq ´ 1 ` εtrpBq ` Opε2q, so Jacpfq “ tr, and the tangent space is

slpn, Cq “ t traceless n ˆ n matrices u

29 Lecture 11 July 14, 2015

Tensor Madness Between Tom Garrity and Sean McAfee, we’ve already covered most of what I want 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 dual space V ˚ is ˚ V “ t linear f : V Ñ Cu. œ œ If G V , then we also get an action G V ˚ by g ¨ fpxq “ fpg´1 ¨ xq.

Deﬁnition 85. The tensor algebra of a ﬁnite-dimensional vector space V is the space 8 bn TV “ V “ C ‘ V ‘ pV b V q ‘ ... n“0 œ à œ If G V , then we also have an action G TV by acting on the components of a tensor individually. For the d-th tensor power V bd, we have a right-action of Sd by permuting the coordinates. The action of Sd commutes with the action of G.

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

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

Ź d V bd “ SdV ‘ V ‘ p other stuff q . ľ Example 87. For example, Sym2V is the tensors that look like x b x, y b y, and x b y ` y b x, and 2 V is the tensors that looks like x b y ´ y b x.

Definition 88. TheŹsymmetric algebra over a finite-dimensional vector space V is 8 n 2 SympV q “ Sym V “ C ‘ V ‘ Sym V ‘ ... n“0 à This is just the polynomial ring Crx1, . . . , xdim V s. Definition 89. The exterior algebra over a finite-dimensional vector space V is 8 n 2 V “ V “ C ‘ V ‘ V ‘ ... n“0 ľ à ľ ľ

30 Notice that for n ě dim V , n V “ 0. Hence, the exterior algebra remains ﬁnite dimensional even though the symmetric algebra is not ﬁnite dimensional. n Ź Additionally, det P V ˚.

Deﬁnition 90. The Źsuper polynomial algebra over V is

8 d SymdV ‘ V dě2 à ´ ľ ¯ 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,

∆ “ pxi ´ xjq. iăj ź

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

λ1`pn´1q λn´1`1 λn λ1`pn´1q λn´1`1 λn Pλ “ x1 ¨ ¨ ¨ xn´1 xn ` ... ` sgnpσqxσp1q ¨ ¨ ¨ xσpn´1q xσpnq, where the sum is over all permutations σ P Sn. Then the Schur Polynomial is Pλ{P0, and P0 “ ∆.

Algebraic Geometry To motivate this, let’s think about the continuous characters of Up1q. These are maps χ: Up1q Ñ Cˆ, and χpeiθeiψq “ χpeiθqχpeiψq. This tells us that χpeiθq must 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 are k maps χkpgq “ g for k P Z. So all of the representations of Up1q lie inside

ikθ Ce kPZ à We can account for these characters in another way, by considering Up1q Ă Cˆ “ GLp1, Cq. What are the algebraic functions on Cˆ “ GLp1, Cq? These are the Laurent polynomials, because we don’t have to worry about plugging in zero in the denominator. ´1 k Crz, z s “ Cz kPZ à 31 This looks a lot like what we found for Up1q! From this toy example, it makes sense to think about the algebraic functions 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 coordinate 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 on the group.

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

Crx11,x12,...,xnns {xdetpxij q´1y When n “ 2, we get x ,x ,x ,x Cr 11 12 21 22s . {xx11x22´x21x12´1y

The coordinate ring of GLpn, Cq is given by

x ,y 1 Crt ij u s x , ´ {xy detpxij q´1y “ C t iju det

“ ‰ 2 We can’t write down a polynomial equation for GLpn, Cq in n dimensions, because it’s the compliment of the zeros of a polynomial equation. Namely, it’s 2 the compliment of detpxijq “ 0. But we can do it in n `1 dimensions as above.

Lecture 12 July 15, 2015

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 topological manifold. We have a sheaf CM of continuous functions on M. For any open set U Ď M, CM pUq “ t continuous f : U Ñ Ru.

There is another example of a sheaf on M which we call ZM . This is the sheaf of 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 connected components of U) it contains a lot of information. From it, we can ﬁnd all of the Betti Numbers of the manifold. If M is instead a smooth manifold, there is another sheaf contained within 8 CM called CM . This is deﬁned by

8 CM 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

8 CM Ą CM Ą OM Ą ZM .

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

Definition 93. A finitely-generated C-algebra is a C-vector space A equipped with a multiplication operation of vectors such that there are a1, . . . , an P A so that the map Crx1, . . . , xns Ñ A defined by xi ÞÑ ai is surjective. In analogy to the differential geometry setting, we will have a topological space SpecpAq and a sheaf of regular functions OX pUq. There are two theorems of Hilbert that are critical to get us going in algebraic geometry.

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

Theorem 95 (Nullstellensatz). The maximal ideals in Crx1, . . . , xns are the ker- n nels of evaluation maps at points ~a “ pa1, . . . , anq P C , i.e. the kernels of maps ev~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 of A. maxSpecpRq “ t maximal ideals of Au

Remark 97. Let A be a C-algebra with generators a1, . . . , an and map φ: Crx1, . . . , xns Ñ A. Then if ker φ “ xf1, . . . , fmy, we know that C x ,...,x x ,...,x A r 1 ns Cr 1 ns , – {ker φ “ {xf1,...,fmy so ideals of A are in bijection with ideals of Crx1, . . . , xns containing f1, . . . , fm. We identify Crx1, . . . , xns{ ker φ with the variety Vpker φq corresponding to the ideal ker φ. In particular, the set of these fi can be thought of as polynomials deﬁning the variety Vpker φq, which has coordinate ring A. This gives a bijection maxSpecpAq ÐÑ Vpf1, . . . , fmq. What’s the topology on our space?

33 Deﬁnition 98. The Zariski topology on maxSpecpAq is the topology with the closed sets ZpIq “ tm | I Ă mu Ă maxSpecpAq.

Joke 99. The Zariski topology is like Texas. There is nothing small in the Zariski 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

Ufi “ maxSpecpAqzZpIq. i“1 ď An application of the basis theorem says that if we have an increasing chain of 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 closed sets. They must stabilize

ZpI1q Ě ZpI2q Ě ... Ě ZpInq “ ZpIn`1q “ ... “ ZpIiq. i č Deﬁnition 101. (a) A topological space in which every nested sequence of closed sets stabilizes is called Noetherian;

(b) a Noetherian topological space X 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 of this was a sheaf of functions on the manifold. So here we’re going to deﬁne the sheaf of regular functions on a domain A. For any domain A, we can form the ﬁeld of fractions

a pAq “ 1 | a , a P A, a ‰ 0 C a 1 2 2 " 2 * 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 of some variety, and so the elements of A are functions on that variety. Hence, we can imagine taking a polynomial f modulo a maximal ideal m as evaluation of f at the point corresponding to m. We call elements a P A regular functions on maxSpecpAq. Any element a φ “ b P CpAq also defines a function on maxSpecpAq as a apmod mq φpmq “ pmq “ , b bpmod mq wherever b pmod mq ‰ 0. For each φ P CpAq, we write the domain of of definition as those points where b pmod mq is nonzero. a U “ m P X | φ “ with bpmq ‰ 0 . φ b ! ) Definition 104. The sheaf of regular functions on X “ maxSpecpAq is the sheaf OX defined by

OX pUq “ tφ P CpXq | U Ď Uφu .

Fact 105. OX pXq “ A.

´1 A consequence of this fact is also that OX pUf q “ Arf s. 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 we have a continuous map f : Y “ maxSpecpBq Ñ X “ maxSpecpAq that comes from h. This also gives a map

˚ ´1 f : OX pUq Ñ OY pf pUqq for each U Ă X “ maxSpecpAq open. These data give a morphism of afﬁne varieties. So this category is indeed quite familiar! It’s the opposite of the category of ﬁnitely-generated C-algebras.

35 Lecture 13 July 16, 2015

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

Euclidean Geometry Local Global gluing Open Balls ÝÝÝÑ topological space zero of systems of equations œ spaces of orbits for G M

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

Algebraic Geometry Local Global Afﬁne Variety Projective Varieties gluing via SpecpAq ÝÝÝÝÝÝÑ ProjpAq regular fns Zariski Topology Spaces of Orbits OX OX

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

Deﬁnition 106. A Cˆ-linearlized C-algebra A is an action of Cˆ on A (written λpxq 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 on ˆ which C acts) generates A as a C-algebra, and in addition A1 is ﬁnite dimensional.

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

A “ C ‘ A1 ‘ A2 ‘ ...

k where Ai “ tx P A | λpxq “ λ xu.

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

8 Crx1, . . . , xns “ Crx1, . . . , xnsdeg“m m“0 à ˆ Then the action of C on the generators λpxiq “ λxi is ordinary multiplication. The space A1 “ Crx1, . . . , xnsdeg“1 is the space corresponding to k “ 1, and generates A (because it includes all xi).

Deﬁnition 108. The irrelevant ideal of a graded ring

8 A “ Ai m“0 à is the ideal 8 m “ Ai. mą0 à This is the maximal Cˆ invariant ideal. Deﬁnition 109.

ˆ maxProjpAq “ t maximal C -invariant homogeneous idealsmx Ĺ mu

Similarly, we can put a Zariski topology on maxProjpAq by declaring the closed sets to be ZpIq “ tmX | I Ă mxu for Cˆ invariant ideals I.

Example 110. If A “ Crx1, . . . , xns, we know that SpecpAq is geometrically Cn. The points in the space maxProjpAq correspond to the lines through the n origin in C , 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 line in projective space. We throw it out much like we throw out the origin when deﬁning projective space.

So now we need to know the functions on this space. We deﬁne the rational functions on maxProjpAq by ratios of homogeneous polynomials of the same degree. F pAq “ F,G P A for some m . C 0 G m " ˇ * ˇ These indeed deﬁne functions onˇ the space, but I leave that as an exercise. ˇ

37 Problem 111. Show that φpmr~psq “ φpmod mr~psq makes sense as a complex number.

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

OX pUq “ tφ P CpAq0 | U Ď Uφu.

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

F Ara´1s “ F P A Ă pAq . 0 ad d C 0 " ˇ * ˇ ˇ Hartshorne calls this Apaq. Note that Cˇ pAq0 is the same as the ﬁeld of fractions ´1 of CpAra s0q. ´1 The afﬁne variety corresponding to this algebra is maxSpecpAra s0q, which is 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. ˆ Given a C -linearized C-algebra with A1 generated by a1, . . . , an as a C- vector space, we have a surjective map CrA1sA. We can think of CrA1s as 1 lines through the origin in the dual vector-space of A1, which we call P pA1q. 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 is locally affine. In affine space, it’s easy to go back from the variety to the algebra 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. So it’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 Veronese embedding. We have to rescale by setting µ “ λd. ˆ If A and B are both C -linearized C-algebras, what space does A bC B determine? It’s the product of maxProjpAq and maxProjpBq.

C ‘ A1 b B1 ‘ A2 b B2 ‘ ...

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

C ‘ xdetpXI1 qy ‘ xdetpXI1 qy ‘ ... where Ii are all size m subsets of t1, 2, . . . , nu.

Lecture 14 July 17, 2015

Today I want to talk about ﬂag manifolds and (hopefully) introduce quiver varieties so you can see what the big kids in the other room (graduate student seminar) are talking about. Let’s talk about the characteristic polynomial. Tom already talked about these things, but I’m going to talk about them again. This feeds right into the representation 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 this as n n´1 n detpxI ´ Aq “ x ´ p1pAqx ` ... ` p´1q pnpAq, for some polynomials pi in the coordinates of A. This is the ring of invariants of this action.

SLpn, q Cry1, . . . , yns “ Crtaijus C

yk ÞÑ pkpaijq

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

q : maxSpecpCrtaijusq Ñ maxSpecpCry1, . . . , ynsq

xxij ´ aijy ÞÑ xyk ´ pkpaijy

But we can think of this as sending a point (maximal ideal) associated to the matrix 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 a matrix in SLpn, Cq whose coefﬁcients of the characteristic polynomial are given by these coordinates.

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

1 ˚ N “ upper triangular matrices with 1 on diagonals 0 1 " „ * Each of these is has the identity matrix in the closure of its orbit under conjugation by SLpn, Cq. Example 114.

´1 2 t 0 1 a t 0 1 at t 0 1 0 “ ÝÝÝÑÑ 0 t´1 0 a 0 t 0 1 0 1 „  „  „  „  „  Let’s try to tie all of this back to representation theory. Given a ﬁnite dimensional 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 difference between conjugation by N and conjugation by the identity, then all of the irreducible representations of SLpn, Cq lie inside the invariant ring of the regular representation. N CrSLpn, Cqs œ Now let’s consider SLpn, Cq CrSLpn, CqsN . This is related closely to the flag varieties we talked about earlier. Recall that the flag variety Flpnq is the set of flags in Cn. n Flpnq “ tV1 Ă V2 Ă ... Ă Vn´1 Ă C u 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 stabilizer of the standard flag? It has to fix subspaces xe1, . . . , eky, so the matrix needs to be upper triangular. Definition 115. The Borel subgroup of a Lie group is the stabilizer of a flag. In our case, this is upper triangular matrices

˚ ˚ ... ˚ 0 ˚ ... ˚ $ , B “ ». . .fi ’ . .. . / &’— ffi./ —0 0 ... ˚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 lecture, and we have an action of T on A.

Example 116. For n “ 2, T “ Cˆ, and

A “ C ‘ A1 ‘ A2 ‘ ...

We also have that 1 maxProjpAq “ Flp2q “ P C. This makes sense, because these things are just ﬂags in C2 (so lines through the origin)! The Borel-Weil theorem says that this is all of the representations of SLp2, Cq.

Example 117. In general, we have an action of T “ pCˆqn´1 on CrSLpn, CqsN . Remark 118.

N SLpn, q T maxSpec CrSLpn, Cqs – C {N ÝÝÝÝÑ 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 variety Flpnq.

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 the quiver a vector space and to each arrow a directed graph. In addition, these arrows 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 these vector spaces in a quiver representation, the maps become matrices. We can act on these spaces by change of basis via

ˆ GLpV1q ˆ GLpV2q ˆ ¨ ¨ ¨ ˆ GLpVnq{C

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

pn qˆ¨¨¨ˆ pn q Spec CrQsSL 1 SL k ´ ¯ These are just some things that are big in algebraic geometry right now. Flag varieties, quiver varieties, Grassmannians, etc.