AROUND THE HORN CONJECTURE

L MANIVEL

Abstract We discuss the problem of determining the p ossible sp ectra of a sum of Hermitian

matrices each with known sp ectrum We explain how the Horn conjecture which gives a

complete answer to this question is related with and

representation theory The rst lecture is an introduction to Schubert calculus from which one

direction of Horns conjecture can b e deduced The reverse direction follows from an application

of geometric invariant theory this is treated in the second lecture Finally we explain in the

third lecture how a version of Horns problem for sp ecial unitary matrices is related to the

quantum of

Notes for Gael VI I I CIRM March

Eigenvalues of hermitian matrices and Schubert calculus

The problem Let A B C b e complex Hermitian n by n matrices Denote the set of eigenvalues

or sp ectrum of A by A A A and similarly by B and C the sp ectra

n

of B and C The main theme of these notes is the following question

Suppose that A B C What can be their spectra A B C

There are obvious relations like trace C trace A trace B or C A B But

a complete answer to this longstanding question was given only recently and combines works

and ideas from representation theory symplectic and algebraic geometry

Weyls inequalities There are various characterizations of the eigenvalues of Hermitian ma

trices many of which are variants of the minimax principle Let j b e the standard Hermitian

n

pro duct on C If s n r denote by G the of r dimensional linear subspaces

rs

n

of C Then

Axjx A min max

j

LG xL

n jj

xjx

n

The idea is to test the values of Axjx on subspaces of C Thats how Hermann Weyl

proved in the following inequalities

Prop osition C A B

pq p q

Proof The rst p oint is to understand what happ ens when you mo dify a Hermitian matrix by

n

another one of small rank To x notations let e e b e a basis of C made of eigenvectors

n

of A for the eigenvalues A A

n

Lemma Suppose that rank B k Then C A

k

Proof For reasons of dimensions the vector space generated by e e meets the kernel

k

of B along a nonzero unitary vector x Then C C xjx Axjx A

k

After adding if necessary suitable multiples of the identity we can supp ose that the eigenvalues

p n

of A B C are all p ositive Denote by A the Hermitian endomorphism of C dened by

p

A e e if i p zero if i p it dep ends on the choice of the basis of eigenvectors

i i

p

when A A The largest eigenvalue of A A A is A A Let

p p p

q p q p q

B B B and C A B C A B Since A B has rank at most p q

the lemma implies that

C C A B A B

pq p q

L MANIVEL

Note that the main p oint in the key lemma ab ove was to nd a linear subspace in sp ecial

p osition In general it is intuitively clear that the eigenvalues of C A B will dep end on the

relative p osition of the eigenspaces of A and B Thats precisely the kind of information that is

enco ded in Schubert varieties of Grassmannians

n

Schubert varieties Let V denote a complete ag in C that is a sequence of linear subspaces

n

V V V C

i n

where V has dimension i Let b e a partition inscrib ed in a r by s n r rectangle that is

i

a sequence of integers s We dene the Schubert cell

r

V fW G dim W V i for s i j s i g

rs j i i

and the Schubert variety

X V fW G dim W V i i r g

rs

si

i

For example when has a unique non zero part k we get a special Schubert variety

X V fW G W V g

rs

k sk

We will use the notations and X when the reference ag V do es not matter The following

facts are wellknown see eg

The Schubert variety X is a closed subvariety of G dened lo cally by the vanishing of

rs

n n

minors of the comp osite maps W C C V

i

Let W G The sequence dim W V go es from to r increasing at most by one at

rs j

each step So it increases strictly at exactly r values of j which we denote by j s i

i

with a partition inscrib ed in a r by s rectangle This proves that

a

G

rs

r s

Moreover if dim W V i we clearly have s i s i hence

i i

si

i

a

X

In particular we have the incidence relation X X if and only if

n

Chose a basis v v of C adapted to the reference ag ie such that V hv v i

n i i

Then W admits a unique basis of the form

X

x v w v

ij j i

si

i

j si

i

j sk k i

k

r sjj

where i r In particular is ane isomorphic to C where jj

r

Moreover it is easy to check that X is the Zariski closure of

It follows that the Schubert cells dene a complex cellular decomp osition of the Grassmannian

An immediate consequence is that the fundamental classes of the Schubert varieties the Schubert

classes X where is a partition inscrib ed in a r by s rectangle form a basis of the

cohomology with integers co ecients

M

H G Z Z

rs

r s

Note that the Schubert classes do not dep end on the reference ag This is b ecause GLn C acts

transitively on the set of complete ags And it is a general fact that if Y is a subvariety of some

variety X on which a connected top ological group G acts continuously then the fundamental

class of g Y do es not dep end on g G

AROUND THE HORN CONJECTURE

P

k k

q rank H G Z b e the Poincare polynomial of the Grass Exercise Let P G

Z rs q rs

k

mannian Prove that when q is a p ower of a prime P G equals the number of p oints of the

q rs

grassmannian G F over the eld with q elements Deduce from this interpretation that

rs q

r s

q q q

P G

q rs

r s

q q q q

How to multiply Schubert classes Now that we know the additive structure of the coho

mology of the Grassmannian we need to understand its multiplicative structure It was

investigated in detail by mathematicians from the last century in particular from the German

and Italian schools Here are the main remarkable formulas

Duality Let again V b e our reference ag with an adapted basis v We dene the dual ag

V by V hv v i From our explicit descriptions of a prefered basis of an element

ni n

i

of a given Schubert cell we easily deduce the following fact if and are partitions such

that jj jj r s then V and V meet transversely in a unique p oint if

where s s is the complementary partition of and have empty intersection

r

otherwise see p or This implies that the cuppro duct

r s

where is the class of a p oint This means that the basis of the cohomology of the Grassman

r s

nian given by Schubert classes is up to complementarity selfdual relatively to Poincare duality

Usually one represents a partition by its diagram as b elow this diagram has b oxes of the

i

ith row from top to b ottom and the complementary partition has complementary diagram

after rotation in the r by s rectangle

r r

s s

Pieris formula Let b e the class of a special Schubert variety Then

k

X

k

r s

k

where k denotes the space of partitions such that j j jj k and see

i i i

p or

Giambellis formula It is a formal consequence of Pieris formula that each Schubert class can

b e expressed as a determinant in sp ecial classes see p or

det

ij ij r

i

n

Exercise Identifying C with its dual vector space there is a natural isomorphism G G

rs sr

G where Show that this isomorphism exchanges a Schubert variety X G with X

sr rs

is the conjugate partition of Deduce from Giambellis formula that if then

k

k

det

ij s

ij i

L MANIVEL

r

s

s

r

There are other descriptions of the cohomology ring of the Grassmannian as a quotient of

a p olynomial ring Indeed one can deduce from Giambellis formula that the sp ecial classes

P

k s

generate H G Let z z z z One can prove

s rs s

k

k

that is the k th of the dual of the tautological vector bundle on G which has

rs

k

rank r so that for k r p moreover this gives a complete set of relations

k

More precisely

H G Z C h i

rs s r n

In principle the previous formulas are enough to compute the pro duct of any two Schu

b ert classes you just need to use Giambellis formula to express in terms of sp ecial Schubert

classes and then apply Pieris formula r times Of course this is not very satisfactory Little

woo d and Richardson gave in a combinatorial algorithm for computing the multiplicities

in the pro duct c

X

c

can b e non zero only when c The LittlewoodRichardson rule First observe that c

A skewtableau on will b e a way to ll the complement of the diagram of inside

that of by p ositive integers which increase on each column from top to b ottom and do not

decrease on each line from left to right It will have weight if each integer app ears exactly

times in the lling Finally let w w w b e the asso ciated word obtained by reading

i

jj

the numbers in the skewtableau line after line from right to left It is a lattice word if in every

subword w w each integer j app ears at least as often as j see I

i

equals the number of skewtableaux on The LittlewoodRichardson coecient c

of weight whose associated word is a lattice word

Example In computing the pro duct the following skewtableaux do contribute and we

get

There exist several other combinatorial descriptions of Littlewoo dRichardson co ecients A

recently discovered one is due to Knutson and Tao and has b een essential in their pro of of the

saturation conjecture see b elow and for a very clear exp osition of the main ideas Consider

AROUND THE HORN CONJECTURE

a triangular array of vertices n on each side and call rhombus a union of two small triangles

with a common side

A

A

A

A

A

A

A

A A

A A

A A

A A

A A

A

A A A

A A A

A A A

A A A

Denition A hive is a labelling of this triangular array such that for each rhombus the sum

of the labels at the obtuse vertices is greater than or equal to the sum of the labels at the acute

vertices

If is any partition dene j j Knutson and Tao proved

n

equals the number of integral that if j j jj jj the Littlewoo dRichardson co ecient c

hives with b order lab els given by jj For example on the picture ab ove there is

an integral hive with these b order lab els for

Schubert calculus and Hermitian matrices We know enough of Schubert calculus to general

P

A ize Weyls inequalities The inequalities we will obtain involve partial sums A

i I

iI

of eigenvalues of a Hermitian matrix A where I is an increasing sequence i i n

r

To such a sequence we asso ciate the partition I i r i

r

The relation b etween the partial sums A and Schubert varieties is as follows If W G

I rs

n

let C W b e the hermitian pro jection and dene the Rayleigh trace

W

r

X

Au u R W trace A

i i A W

i

n

for any orthonormal basis u u of W Let A b e a complete ag in C compatible with

r

b e the the eigenspaces of A which means that A ha a i with Aa Aa Let A

i i i i i

opp osite ag It is an exercise to check that

R W A max

A I

W A

(I )

Prop osition Let A B C be Hermitian matrices such that A B C Let r n and

I J K be increasing subsequences of length r with associated partitions Suppose that

is non zero Then the LittlewoodRichardson coecient c

A B C

I J K

Proof If c the duality prop erties of Schubert classes implies that the intersection of the

and C must b e non empty Let W b e an intersection B Schubert varieties A

p oint Denote by K the increasing sequence corresp onding to so that k n k we

i r i

have C C Using the linearity of the Rayleigh trace we get

K

K

A B R W R W R W R W C C

I J A B C C K

K

This prop osition due to HelmkeRosenthal and Klyachko see Theorem raises a

number of questions Are these inequalities sucient for the existence of Hermitian matrices

A B C with the corresp onding eigenvalues If it is the case why is it enough to consider

linear inequalities in the eigenvalues And how can we characterize the triplets of

is non zero We partitions such that the corresp onding Littlewoo dRichardson co ecient c

will answer to the rst two questions in the next lecture The answer to the last stems from a

L MANIVEL

unexp ected relation of Littlewoo dRichardson co ecients with the representation theory of the

linear group

Representations of GL(n C ) Let V b e a complex vector space of dimension n The linear

group GLV is linearly reductive which means that any of its nite dimensional rational

representation is a direct sum of irreducible ones ie without any stable prop er subspace

Moreover these irreducible representations are classied by nonincreasing sequences of integers

Among these are the partitions corresp onding to p olynomial represen

n

tations For simplicity let us supp ose that is indeed a partition Then the corresp onding

irreducible representation S V can b e dened in the following way choose any numbering N of

the lenghs of the columns the diagram of by integers from to l jj denote by

m

of this diagram Then we have a comp osite map

V V

l

n

1

m

1

V V V S V S V p

N

V

l

i i

are the usual ones but they involve in V the V V dened as follows the inclusions

factors V in the p ositions prescrib ed by the numbers in the ith column of N the pro jections

l

j j

V S V are the usual symmetrizations but they involve in V the factors V in the

p ositions prescrib ed by the numbers in the j th line of N These maps are obviously compatible

with the diagonal action of GLV thus the image of p is a GLV mo dule this is S V For

N

k

example if has only one non zero part say k then we get the symmetric p ower S V if

V

k

has k non zero parts all equal to one we get the skewsymmetric p ower V

The representation theory of the linear group was developped in the rst decades of the

century in particular by I Schur Schur knew how to decomp ose the tensor pro duct of any

Schur module S V with a symmetric p ower But it was only in that Lesieur realized that

the answer is formally identical with Pieris formula up to the fact that in each case we consider

partitions with slightly dierent restrictions inscrib ed in a r by s rectangle for the Grassmannian

G with at most n parts for GLn C But as we have seen b ecause of Giambellis formula

rs

which is a formal consequence of it Pieris formula is enough to determine the multiplication

of Schubert classes hence also of Schur mo dules In particular the multiplicity of S V inside

the tensor pro duct S V S V is equal to the Littlewoo dRichardson co ecient c

There is a more geometric way to dene Schur mo dules as spaces of sections of line bundles

Consider the variety of partial ags V V V where dim V rep etitions are

m i

i

allowed when has several columns of the same size Using a Plucker embedding for each V

i

V V

m

1

V then a Segre embedding we obtain a subvariety F V of PW where W V

The BorelWeil theorem see then asserts that

F V O S V

Principles of Geometric Invariant Theory

The general problem Let X b e an algebraic variety dened over an algebraically closed eld

k with an action of an ane algebraic group G Can we construct an orbit space for this action

that is an algebraic variety Y with a surjective morphism f X Y such that the b ers of

f are exactly the Gorbits in X The answer is scarcely yes this would imply for example

that all orbits are closed ie the action would b e closed but we can ask for quotients with

weaker prop erties An imp ortant notion is the following

Denition A categorical quotient of X by G is a pair Y f such that f is constant on Gorbits

and such that every morphism g X Z with the same property factors through f

The affine case Let X Sp ecA where G acts rationally on the nitely generated k algebra

A This is a particularly nice case when G is a reductive group in the sense of the rst lecture

if char k in general the denition is dierent indeed it was proved by Weyl for char k

G

and by Nagata in general that the algebra A of Ginvariants is nitely generated Theorem

AROUND THE HORN CONJECTURE

G

One can then dene the ane variety Y Sp ecA and the natural morphism f X Y

has the following nice prop erties Theorem

G

O f O

Y X

the image by f of any closed invariant subset is closed

f separates disjoint closed invariant subsets

In particular two p oints in X have the same image i their orbit closures meet A consequence

of these prop erties is that Y f is a categorical quotient Moreover if U is an op en subset of

Y such that the action of G on f U is closed then U is an orbit space

The projective case For simplicity let us consider the case where V is a Gmo dule and

X PV a Ginvariant closed subvariety

Denition A go o d quotient of X by G is a pair Y f where f is surjective ane constant

on Gorbits and satises properties above It is a geometric quotient if its also an orbit

space

In general such quotients will not exist But they will if we restrict ourselves to suitable op en

subsets of X The rst observation is that we need Ginvariant forms to reduce the problem to

the ane case This motivates the following imp ortant denition for which we follow

Denition A point x X is semistable if there exists a non constant Ginvariant homogeneous

form P on V such that P x It is stable if moreover its stabilizer G is nite and one

x

can nd P as above such that the action of G on X fy X P y g is closed

P

ss s

Denote by X and X the op en subsets of X consisting of semistable and stable p oints

Note that they may very well b e empty

ss

Fundamental theorem There exists a good quotient Y f of X by G and Y is pro jective

s s s s

Moreover there exists an open subset Y of Y such that f Y X and Y f is a

s

geometric quotient of X

The HilbertMumford criterion Except for very simple cases it is extremely dicult from

the denition ab ove to determine which are the stable or semistable p oints of a given action

Indeed this would imply to compute all the Ginvariant p olynomials which is intractable in

general Let x X PV and v V lying over x One can prove Prop osition

x is semistable i the closure of Gv does not contain the origin

x is stable i the morphism from G to V given by g g v is proper

From this it is p ossible to derive a numerical criterion for stability Recall that a one pa

rameter subgroup of G is a homomorphism k G The induced action of k on V can b e

diagonalized we can nd a basis e e of V and integers k k such that

n n

n n

X X

k

i

v e V t v e for t k v tv

i i i i

i i

Dene x minfk v g Note that when t tends to zero tx has a limit x

i i

P

corresp onding to v v e Moreover x x The Hilb ertMumford

i i

k

i

criterion states that a p oint is semistable i it is semistable with resp ect to every one

parameter subgroup Theorem

x is semistable i x for every one parameter subgroup of G

Application to flags We will apply the Hilb ertMumford criterion on two examples the rst

b eing a simple version of the second one

Let U E b e vector spaces The action of G SLU on W U E induces for each

V V

r r

integer r an action on W and on the Grassmannian G r W P W

L MANIVEL

Prop osition A point L G r W is semistable i for every proper subspace V of U the

subspace M L V E of W is such that

dim L dim M

L M

dim V dim U

Proof Let b e a one parameter subgroup of G we choose a basis u u of U such that

n

r

i

tu t u with r r and let U hu u i and L L U E Let

i i n i i i i

L L L b e those L L and denote their dimensions by l l Cho ose an

p p i i m

m

1

P

adapted basis f of L If l j l we have f u v for some v V Then

j i i j

k k j k j

k p

i

L lim tL is the space generated by the u v and a simple computation shows

t p p j

i i

that

n

X X

r dim L L L L r dim L L

i i i p p p

j j j 1

i j

If the criterion given by the prop osition is fullled then dim L imn and using the identity

i

r r which follows from the fact that is a subgroup of SLU we get L

n

Hence L is semistable The reverse statement is an exercise

i

of V the dimensions Let now V b e a mltration of V that is a family of m ltrations V

of each subspace is xed Each such ltration denes a p oint of some ag manifold and through

a Segre pro duct a mltration thus denes a p oint in the pro jectivization of a tensor pro duct

of wedge p owers of V or of Schur mo dules which is endowed with a natural SLV action

Applying the Hilb ertMumford criterion as in the pro of of the previous prop osition we nd

Prop osition A mltration V is semistable i for every proper subspace L of V

X

i

dim L V V L

j

dim L

ij

Hermitian matrices again We have seen in our rst lecture that each non zero Littlewoo d

Richardson co ecient gives restrictions on the set of eigenvalues of triplets of Hermitian matrices

with sum zero We now want to prove the reverse statement and its obvious extension to a

greater number of matrices This is due to A Klyachko

Prop osition There exists Hermitian matrices A Am of size n having for spectra

the weakly decreasing sequences m and such that A Am is scalar i

m m n

X X X

k k

i

I k

r n

i

k k

for al l r n and al l mtuples I I m of increasing sequences of r positive integers such

that the product of the corresponding Schubert classes is non zero

Proof The necessity of these conditions is checked as in Prop osition To prove the reverse

statement we can use a density argument to make a few additional assumptions rst we

supp ose that the sp ectra i are strictly decreasing rational sequences that they are p ositive

after adding if necessary suitable scalar op erators and even after multiplying by some integer

that they are partitions with distinct parts second we supp ose that all the inequalities ab ove

are strict

i

n

in C from which we construct a mltration by letting Then we choose generic ags F

i

i

where i p i V F

p p

ip

i

We let Ai b e the sum of the Hermitian pro jections on the V which has the required sp ectrum

p

There remains to prove that A Am must b e scalar

AROUND THE HORN CONJECTURE

For this we choose a Hermitian metric on V and for each i a Hermitian basis v i of V

V

i

ip

We consider our mltration as a p oint in PW where W V adapted to the ag F

ip

and a p oint ab ove it is

v v i v i i p W

ip

Because of the previous prop osition and the hypothesis our mltration is stable hence the

SLV orbit of v is closed and do es not contain the origin There is therefore a p oint in this

orbit say v itself which minimizes the distance to the origin for the induced norm on W This

implies that for all X E ndV with trace X we have

X X

i

ReX v v Re R F Re trace XAi

X

ip

ip i

Since the Ai are Hermitian this implies our claim

Corollary Let A B C be weakly decreasing sequences of n real numbers such that

P P P

C Suppose that for every r n and every increasing sequences B A

i i i

i i i

I J K of length r with associated partitions such that c one has

A B C

I J K

Then there exists Hermitian matrices A B C of size n with spectra A B C

The saturation conjecture By the very denition a mltration is semistable if and only if

there exists a SLV invariant form on W which do es not vanish at the corresp onding p oint in

the pro duct X of ag varieties This pro duct has a natural pro jective embedding use a Plucker

embedding for each subspace and comp ose with a Segre embedding corresp onding to a very

ample line bundle O By the BorelWeil theorem we have

X O N S V S V

N A N Am

If the condition of Prop osition are fullled there exists a semistable p oint in X and this

implies that for some N the ab ove tensor pro duct must contain some trivial factor trivial

as a representation of SLV

A priori one has no control on the integer N but the saturation theorem of Knutson and

Tao says that we can always take N

Theorem There is an integer N such that S V S V contains a trivial

N N m

factor i S V S V itself contains a trivial factor

m

We will not explain the pro of of this theorem which follows from a careful study of the com

binatorics of hives Let us simply notice that for m S V S V S V contains a

for trivial SLV factor i jj j j v w for some integer w where v dim V and c

the complementary partition of in the v by w rectangle A consequence of all this is that

the nonvanishing of Littlewoo dRichardson co ecients can b e checked recursively

Horns conjecture We can now state the original conjecture of Horn which go es back to

and is also recursive in nature Dene the sets of triples of increasing sequences of r integers

in f ng

r r

n

g fI J K jI j jJ j jK j U

r

pp

r n n

g p r F G H T jI j jJ j jK j fI J K U T

F G H

p r r

Theorem There exists Hermitian matrices A B C of size n such that A B C i

trace A trace B trace C and

n

A B C r n I J K T

I J K r

L MANIVEL

Proof One checks using induction that this is just a reformulation of Prop osition and Corol

lary For more details see Theorem

For n we get For n we obtain twelve inequalities

Why polytopes There seems to b e no reason a priori why the set of eigenvalues of Hermitian

matrices whose sum is zero should b e describ ed by linear inequalities that is should b e a convex

polytope It turns out that such p olytop es do app ear in the general context of torus actions on

symplectic varieties of which our problem is a sp ecial case

Denition Let M be a manifold with an action of a connected Lie group K preserving a

symplectic form Dierentiating this action we associate to each X k the Lie algebra of

K a vector eld on M A map M k is then a moment map for the action of K

X

in M if it is K equivariant with respect to the coadjoint action of K on the dual k of its Lie

algebra and for al l X k dX equality of forms on M

X

This is equivalent to the existence of a Hamiltonian H k O M the space of regular

functions on M which is a Lie algebra homomorphism and lifts the natural map k T M

induced by the action Here T M is the space of vector elds on M if f O M its dierential

df can b e identied via the symplectic form with a vector eld The moment map and the

f

Hamiltonian are related by the identity H X m mX for m M X k

Moment maps have go o d functorial prop erties if N is a K invariant submanifold of M the

restriction is a moment map for the restricted action Also if we restrict the action to a

jN

subgroup L of K we get a moment map for this new action by comp osing with the pro jection

k l

n

Example The action of the unitary group U n on P endowed with the symplectic

structure given by the FubiniStudy metric is Hamiltonian the moment map asso ciates to each

n n

p oint of P the Hermitian pro jection on the corresp onding line in C Therefore the action

n

of any subgroup K of U n preserving a subvariety X of P is also Hamiltonian

Example Let M b e any coadjoint orbit in k Let k b e the stabilizer of a p oint m M

m

Then T M kk and the identity X Y hX Y mi denes a symplectic form on M

m m

One checks that the inclusion M k is a moment map for the action of K on M

Example Take in particular K U n so that k is the space of skewHermitian matrices

Then k can b e identied in a K equivariant way with the space H of Hermitian matrices via

n

the trace form H trace iH This identies the coadjoint orbits in k with the K orbits

in H which are just the spaces O of Hermitian matrices with xed sp ectrum a weakly

n

decreasing sequence of real numbers

The following theorem is due to Atiyah and GuilleminSternberg see

Theorem Let M be a compact connected Suppose that a torus T acts

on M with a moment map u M t Then uM is a convex polytope More precisely the

image under u of the xed point set of T in M is nite and uM is the convex hul l of this

nite set

In the example ab ove we can restrict the action on O to the subgroup T of K consisting

of diagonal matrices which is a torus group The induced moment map u O t takes a

Hermitian matrices to its diagonal entries The theorem then asserts that the diagonal entries of

the matrices with sp ectrum describ e the convex hull of the p ermutations of This is known

as the SchurHorn theorem

AROUND THE HORN CONJECTURE

For a Hamiltonian action of a Lie group which is not a torus group the image of the moment

map needs not b e convex but still a part of it must b e convex Supp ose that K is compact let

T b e a maximal torus in K and t t a p ositive Weyl chamber when K U n the main

can b e chosen to case of interest to us T is the torus group of diagonal matrices in K and t

b e the cone of weakly decreasing sequences

Theorem Let M be a compact connected symplectic manifold with a Hamiltonian action

of K Then uM t is a convex polytope

To state this theorem which is due to F Kirwan Theorem in a slightly dierent

which for K U n takes a Hermitian matrix to its way consider the map p k t

sp ectrum Then p uM t is a convex p olytop e This holds for any compact group the

at a single p oint map p b eing dened through the prop erty that each K orbit in k meets t

Example Consider the diagonal action of K U n on O O This action is Hamiltonian

its moment map takes a pair of Hermitian matrices to their sum Comp osing with the map

p ab ove we get that the sp ectrum of the sum of two Hermitian matricies with given sp ectra

describ es a convex p olytop e This justies qualitatively the Horn conjecture See for a

quantitative discussion along the same lines

The quantum case

The multiplicative problem Let A SU n Its eigenvalues are complex numbers of norm

one which we can write in a unique way as expi with and

i n

These inequalities dene the fundamental alcove U and we denote by

n

A U the sp ectrum of A The multiplicative analogue of Horns problem is then how can

we describ e the set

l fA A A A SU n A A I g

q

l l l

A geometric interpretation For U denote by O the space of sp ecial unitary matrices

with sp ectrum Consider the op en curve P minus l p oints p p its fundamental group

l

is generated by l small lo ops around p p with the single relation

l l l

Thererefore there is an identication b etween

N fA A O O A A I gS U n

l l l

1

l

and the mo duli space M of unitary ndimensional representations up to global con

l

jugacy of P fp p g such that the image of is in O In particular

i

l l

i

l i M is non empty

q

l

There is another interpretation of this mo duli space due to Mehta and Seshadri This involves

the concept of stable vector bund les which is of course closely related with the stability concept

of Geometric Invariant Theory see Chapter On a smo oth complete curve C a vector

bundle E is semistable if for every prop er subbundle F of E we have

deg F

E F

rankF

Supp ose that the genus of C is at least two and that C H where H denotes the Poincare

halfplane and C is a discrete subgroup of AutH PSL R acting freely on H Then

it is a classical theorem of Narasimhan and Seshadri that the space of isomorphism classes of

unitary representations of is in bijection with the mo duli space of semistable vector bundles of

degree zero on C The denition of this corresp ondance is the following to a representation

n

U n one asso ciates the vector bundle E H C on C

The theorem of Mehta and Seshadri is an extension of this result to noncompact Riemann

surfaces with nite volume Such a surface can b e seen as the complement of a nite set in a

complete curve P in the case of interest to us

L MANIVEL

Denition A parab olic vector bundle E on P is a vector bund le of rank n plus additional

data a complete ag E of subspaces of each ber E weights which are real numbers

p p ij

i i

such that Its parab olic degree is

i in i

X

pardeg E deg E

ij

ij

Actually the weights will only play a role in the denition of semistability Consider a sub

bundle F of E It can b e endowed with a parab olic structure in the following way each b er

F has an induced complete ag given by taking the distinct terms in the sequence F E

p p p j

i i i

and we asso ciate to each of these subspaces the maximum of the corresp onding weights Then

E is semistable i

pardeg F

E F E F

p p

rankF

Theorem Suppose that are rational sequences Then the space M is

l l

homeomorphic to the moduli space of semistable parabolic vector bund les on P of degree zero

and parabolic weights at p

ij i

Moreover one can show that the generic p oint E of this mo duli space must b e semistable

as an ordinary vector bundle Lemma On P a vector bundle is a direct sum of line

bundles and is semistable i all these line bundles have the same degree Here E has degree

zero and is therefore trivial

The quantum cohomology of the Grassmannian A subbundle F of the trivial bundle E is

simply given by a map P G such that F S where S is the tautological vector

F rs

F

bundle on G Since det F O we have

rs

F

X X

pardeg F deg

j F

l j

j I j I

1

F F

l

with resp ect to the where the sequence I enco des the relative p osition of p E

i F F i p

i

complete ag E We get

p

i

l

Theorem The set l U is the polytope dened by the inequalities

q

X X

A d A

j j

l

j I j I

1

l

where I I are increasing sequences of r n elements in f ng such that there exists a

l

degree d map P G sending p p to Schubert cells in general position

rs I I

l

1

l

Now the question is to nd necessary and sucient conditions for such maps to exist A

ma jor discovery of the last ten years has b een that these maps can b e used to construct cer

tain q deformations of cohomology rings whose very striking prop erties have had sp ectacular

applications sp ecially in This is the theory of quantum cohomology for

which we refer to

We will b e primarily interested in the smal l quantum cohomology ring QH G of the Grass

mannian G G This ring can b e dened as the space H G C q where q is an indetermi

rs

nate endowed with the following commutative and asso ciative pro duct

X

d

c dq

d

d is dened as follows It is equal to where the quantum LittlewoodRichardson number c

the number of degree d maps P G sending three given p oints to Schubert cells

1

The word parab olic comes from the uniformization theory of noncompact Riemann surfaces

2

In a more general setting one can dene quantum cohomology in terms of GromovWitten invariants which

are dened through mo duli spaces of stable maps from a p ointed curve to a variety with certain prop erties The

miracle is that this pro duct is associative Theorem

AROUND THE HORN CONJECTURE

in general p osition when this number is nite and zero otherwise A dimension count shows

that c d can b e nonzero only when j j nd jj jj as follows from Theorem i

Note that c c is the usual Littlewoo dRichardson co ecient and that the quantum

cohomology ring is a q deformation of the ordinary cohomology In the case of the Grassmannian

one can give a simple description of this ring

P

s s k

Theorem Let z z z z Then

s

k

k

s

QH G QH C q h q i

s r n n

rs

s

Proof We rst check that the relations q do hold in QH G First

r n n

d we must have j j nd jj jj which implies that d when recall that if c

jj jj n upto degree n the quantum and the classical pro ducts coincide Since the

relations do hold in H G see the rst lecture they hold in QH G as well

r n

s

For the remaining relation note that the formal identity

n n s r

s

reduces in QH G to Therefore all we need to check is that c for

n s r

r r

s and s The corresp onding Schubert varieties are resp ectively fW H g

fW l g and fW g where H is some hyperplane and l a is line Note that a line in G must

b e of the form fA W B g the dimensions of A and B b eing r and r In general

there is a unique such line meeting the three Schubert varieties ab ove given by A W H

and B hW l i and our claim follows

This is enough to prove the theorem indeed we proved that there is a ring homomorphism

QH G which is an isomorphism mo dulo q But these two Zq mo dules are free of QH

rs

the same rank hence they must b e isomorphic

Corollary The quantum cohomology ring of the projective space is

n n

QH P Zt q ht q i

Quantum schubert calculus What do es remain of the formulas of Pieri and Giambelli in

quantum cohomology A quite surprising result due to A Bertram is that Giambellis

formula holds without any quantum correction

Prop osition For every partition inscribed in the r by s rectangle one has in QH G

det

ij r

ij

i

The pro of uses a generalization of Giambellis formula due to Kempf and Laksov which applies

to the relative situation where one has a morphism u E F b etween vector bundles on

some variety X and a ag of subbundles of E One then lo oks at the p oints of X ab ove which

the kernel of u has a given relative p osition with resp ect to these subbundles Under genericity

assumptions this denes a subvariety of X whose fundamental class is given by a Giambelli type

formula in terms of Chern classes of E and F see Such a description can precisely

b e used in the context of quantum cohomology by dening generalized Schubert varieties on

the socalled Quotscheme and the KempfLaksov formula proves the prop osition

For any partition we can dene the class det in QH G

l ij ij l

i

s We will give an This class is obviously zero if r but not necessarily when

algorithm to express such classes in terms of those corresp onding to partitions contained in the

r by s rectangle and show how this leads to a formula for quantum Littlewoo dRichardson

co ecients

Denition Choose a box on the border of the diagram of and make n steps on this border

in the northeast direction to cover a nrim of This nrim is legal if its complement in is

again a partition illegal otherwise The width of a nrim is the number of columns it occupies

In the picture b elow you see an illegal and a legal rim in the partition

L MANIVEL

Lemma If contains an il legal nrim or if and contains no nrim then

r

r w

If contains a legal nrim of width w and complement then q

r

Proof The key observation is that the relation q implies by induction that for

n

r

If contains no all j q Let we have det

nj j ij

ij

1 i

nrim then n and if moreover which means that r then the rst line

r

of this determinant is identically zero

Now consider some nrim of b eginning on column a and ending in column b This nrim is

r

illegal precisely when a n b Using the relations q on

a nj j

b

the ath line of the determinant ab ove we obtain a new line that is prop ortional to the b th

line hence

If the nrim is legal we pass this new line to the bth row to obtain up to the sign a

determinant of the same kind which is precisely

This lemma gives the algorithm we were lo oking for b eginning with a partition one can

remove legal nrims until there is no more in the remaining partition This partition do es not

dep end on the choice of the nrims removed it is known as the ncore of Nor do es the sign

mr w

where w is the sum of the widths of the nrims removed and m is their

number see Th Then if is not contained in the r by s rectangle and

otherwise

m

q

Example The partition has for core and the pictures b elow show dierent

ways of removing legal rims

We are now able to compute quantum Littlewoo dRichardson numbers

Prop osition The quantum LittlewoodRichardson numbers can be expressed in terms of

ordinary LittlewoodRichardson numbers in the fol lowing way

X

c d c

where the sum is over partitions with r that can be obtained from by adding d nrims

but needs not be contained inside the r by s rectangle

Proof If we forget the relations the Littlewoo dRichardson rule tells that

s n

X

c in C

s

In QH G we express the right hand side in terms of partitions inscrib ed in the r by s rectangle

and the algorithm ab ove gives the claim

Example Let and let us compute c for r s

n We rst add a rim there are only two such rims that can contribute corresp onding

to and Applying the Littlewoo dRichardson rule one checks

hence c that c and c

AROUND THE HORN CONJECTURE

Exercise Check that QH G has the following multiplication table

q

q q q

q q q

q q q q q

q q q q q

Exercise Deduce from the previous prop osition the quantum Pieri formula

X X

q

k

r s r s

k k

where the quantum contribution k is the set of partitions of size jj jj k n such

that

s s

This is enough in principle to list a complete set of conditions in Theorem Indeed if

their exists a map P G of degree d hiting a collection of Schubert cells in general p osition

one can show that for some e d there exists a nite nonzero number of maps of degree e with

the same prop erty Lemma This means that the corresp onding quantum Littlewoo d

Richardson co ecient is nonzero and this can b e checked with the help of Prop osition

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