THE SYMPLECTIC AND ALGEBRAIC GEOMETRY OF HORN’S PROBLEM

arXiv:math/9911088v2 [math.RA] 25 May 2000

ALLEN KNUTSON

A BSTRACT. One version of Horn’s problem asks for which λ, µ, ν does Hλ + Hµ + Hν = 0
have solutions, where Hλ,µ,ν are Hermitian matrices with spectra λ, µ, ν. This turns out tobe a moment map condition in Hamiltonian geometry. Many of the results around Horn’s
problem proven with great effort “by hand” are in fact simple consequences of the modern
machinery of symplectic geometry, and the subtler ones provable via the connection togeometric invariant theory. We give an overview of this theory (which was not available toHorn), including all definitions, and how it can be used in linear algebra.

1. I NTRODUCTION
This is an expository paper on the symplectic and algebraic geometry implicit in Horn’s
problem, which asks the possible spectra of a sum of two Hermitian matrices each withknown spectrum.
The connection with symplectic geometry is very straightforward: the map (Hλ, Hµ) 7→
Hλ + Hµ that takes a pair of Hermitian matrices with known spectra to their sum is a
moment map for the diagonal conjugation action of U(n) on a certain symplectic manifold(definitions to follow). This is a very restrictive property of maps, and many things can beproved about them. The proofs, at heart, are not really any different than the techniques
Horn himself used to study this map. Nonetheless the framework is worth understanding
in order to recognize what other linear algebra problems are likely to have answers as niceas the ones to Horn’s problem. In particular the Schur-Horn theorem follows very easily
from the general theorems in this area (and was a primary inspiration for them).
Some of the more esoteric connections – to algebraic geometry and representation theory – can also be seen in this context, via the Kirwan/Ness theorem (which we will also
state). Again, the basic techniques used are the same, but in the Kirwan/Ness theorem
one sees these techniques pushed to prove the statements in what appears to be theirproper generality.
Along the way we explain the relation between Hermitian matrices, flag manifolds,
and the Borel-Weil-Bott-Kostant theorem.

2. T HE S CHUR -H ORN THEOREM , H ORN ’ S PROBLEM , AND H AMILTONIAN MANIFOLDSLet λ = (λ1 ≥ λ2 ≥ . . . ≥ λn) be a weakly decreasing list of real numbers, which we’ll
use to encode the eigenvalue spectrum of a Hermitian matrix. The Schur-Horn theorem
states the following:
Date: February 1, 2008.
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Theorem. Let Oλ be the space of Hermitian matrices with spectrum λ. Let Φ : Oλ → Rn take amatrix to its diagonal entries. Then the image of Φ is a convex polytope, whose vertices are the n!
permutations of λ.
This theorem was interpreted by Kostant in 1970 as the U(n) case of a theorem forarbitrary compact Lie groups, leading the way to a much wider generalization found in1982 by Atiyah and independently by Guillemin and Sternberg:
Theorem. Let M be a compact connected symplectic manifold, with an action of a torus T . LetΦ : M → t∗ be a moment map for this action. Then the image of Φ is a convex polytope, theconvex hull of the images of the T -fixed points on M.
Of course, to see how to cast the Schur-Horn theorem in this formulation, we’ll needto define “symplectic manifold” and “moment map”. We will make no attempt to beencyclopædic in our references and instead direct the reader to• [GLS] – for all matters symplectic or convex• [F] – for the algebraic geometry of flag manifolds• [MFK] – for geometric invariant theory• chapter 8 of [MFK] – for the Kirwan/Ness theorem
and references therein.
2.1. Symplectic manifolds. Let M be a manifold, and ω an antisymmetric inner producton the tangent spaces to M, sort of a skew Riemannian metric. Since the inner product ofany vector with itself is zero, we can’t talk about positive definiteness, and so we instead
ask for nondegeneracy – that for any tangent vector ~v1, there exists another vector ~v2 with
ω(~v1,~v2) nonzero. Surprisingly, this forces M to be even dimensional.
There is a standard example: R2d with basis {~x1, . . . , ~xd, ~y1, . . . , ~yd}, where ω(~xi, ~xj) =ω(~yi, ~yj) = 0, but ω(~xi, ~yj) = −ω(~yj, ~xi) = δij (Kronecker delta).
If (M, ω) is locally isomorphic to R2d with its standard ω, we say that M is a symplectic manifold, and call ω its symplectic form.1 This is roughly analogous to studying
Riemannian manifolds that are locally Euclidean.
As with a Riemannian metric, we can talk about the symplectic gradient Xf of a function f, also called the Hamiltonian vector field Xf associated to a Hamiltonian f. It isdefined uniquely by the equation
D~v(m)f = ω(~v(m), Xf(m))
where ~v(m), Xf(m) are tangent vectors to the point m ∈ M, and D~v(m)f is the directional
derivative of f in the direction ~v(m). That this defines Xf uniquely follows from ω’snondegeneracy.
It is easy to show that symplectic gradients have an immense advantage over Riemannian gradients: the derivative of ω along Xf vanishes. In integral form this says that thetime t flow map from M to M, given by following Xf for time t, takes ω to itself.
Example 1. Let M = S2, and ω be the area form, taking a pair of vectors to the oriented
area of the parallelogram they define. Let f be the height function on M, normalized totake the North pole to 1 and the South pole to −1. Then the Riemannian gradient points
1

The usual definition is in terms of de Rham forms, and has this formulation as Darboux’s theorem.
2

down everywhere along longitude lines, as in figure 1, whereas the symplectic gradientpoints sideways everywhere along latitude lines – and generates rotations, which are ofcourse area-preserving.

F IGURE 1. The Riemannian gradient vs. the symplectic gradient of theheight function on S2.
This gives us a handy way to describe (certain) flows on a symplectic manifold – interms of functions, which opens up the world of e.g. Morse theory. One example of thisconnection: a critical point of a function f is one where all the directional derivatives arezero, which (by ω’s nondegeneracy) is equivalent to the vanishing there of Xf. So thecritical points of f are exactly the fixed points of the flow generated by its symplectic
gradient Xf.
2.2. Moment maps. Now let M be a symplectic manifold with symplectic form ω, andan action of a connected Lie group K; for us K will always be the circle group S1, a productT of circle groups, or the unitary group U(n).
If we assume K acts smoothly on M, the elements of K nearby the identity give us
diffeomorphisms of M very close to the identity; differentiating this picture, we find thateach tangent vector to the identity of K gives a vector field on M. Denote the tangent
space to the identity, also known as the Lie algebra of K, by k, and its dual by k∗ . (Thisfunny letter is a Fraktur k.)
Since K acts on itself by conjugation, fixing the identity, it acts on k; this is called the
adjoint representation, and the induced action on k∗ the coadjoint representation.
We say that a map Φ : M → k∗ is a moment map for the action of K on M if

1. Φ is equivariant, i.e. ∀k ∈ K, m ∈ M, Φ(k · m) = kΦ(m)k−1
2. for each ~k ∈ k, the vector field induced on M by ~k equals the symplectic gradient ofh~k, Φi. (We are using here the natural pairing between k and k∗ .)Note that for the action to have a moment map at all, the action of K on M must preserve
the symplectic structure. This is almost sufficient (but not quite); in any case all the actions
considered in this paper will have moment maps. If the action has a moment map it issaid to be Hamiltonian and M is called a Hamiltonian K-manifold.
Since K is connected, we can recover its action from the action of the Lie algebra, which
in turn we can get from the moment map; in particular the moment map uniquely determines the action. (The action does not determine the moment map uniquely even when
one exists, but only up to certain translations.) And while any map Φ : M → k∗ will give
us associated vector fields Xh~k,Φi , very few maps will give actions of K; moment maps arevery special.
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We need four particularly important facts about moment maps. (All follow easily fromthe definition.)
1. Let K act on M with ΦK : M → k∗ a moment map for the action on M, and ρ : H → K
be a Lie group homomorphism, making H act on M too. There is a corresponding
map of Lie algebras dρ : h → k, and a dual map (dρ)∗ : k∗ → h∗ . Then the action of H
is also Hamiltonian, with moment map given by composing ΦK and (dρ)∗. The caseof greatest interest is when H is a subgroup of K, and ρ is inclusion.
2. Let M be a coadjoint orbit, i.e. an orbit of K’s action on k∗ . Then there exists uniquely
a symplectic structure on M such that the inclusion map M ֒→ k∗ is a moment map.
(This was called the Kirillov-Kostant-Souriau symplectic structure until Alan Weinstein found it in 19th-century notebooks of Lie.)
3. Let M be a Hamiltonian K-manifold, and N a Hamiltonian H-manifold. Then thenatural action of K × H on M × N is Hamiltonian, with moment map the direct sumof the two individual moment maps.
4. Let M be a Hamiltonian K-manifold, and N a submanifold of M invariant under K,
such that the restriction of the symplectic form on M is nondegenerate on N (making
N a symplectic manifold in a natural way). Then the action of K on N is also Hamiltonian, with moment map the composition of inclusion with M’s own moment map.
Since the image of a moment map is K-invariant, it must be a union of coadjoint orbits;in this way one can regard the individual coadjoint orbits as sort of “minimal” Hamiltonian K-manifolds. (This is only interesting for nonabelian groups K, insofar as the coadjoint
action is trivial for abelian groups, with the coadjoint orbits just points.)
Example 2. Let M = R2 with the standard symplectic form, and K = S1 acting on M byrotation. Then we can identify K’s tangent space with R, such that the moment map isΦ(~v) = |~v|2. The image of Φ is the positive half-line.
Example 3 (using property 3 above of moment maps). Let M = R2d with the standard symplectic form, and K = (S1)d, each circle acting on a pair of the coordinates – really the nth
power of the previous example. Then we can identify K’s tangent space with Rd. The image of Φ is the positive “orthant” (the d-dimensional generalization of the first quadrantin R2, octant in R3, etc.).
This second example gives a hint of the source of convex polytopes in the Atiyah/GuilleminSternberg theorem stated above; since we’ve defined a symplectic manifold as one thatlooks locally like R2d with the standard form, we just need to prove a “T -equivariant version” of that local normal form, in order to know that at least locally the image of the
moment map is a polytope. The actual proof of A/G-S requires some more ingredients(not explored here) to show the global statement.
Example 4. Let M = CPn = {[z0, z1, . . . , zn] : zi not all 0}/C× , complex projective n-space.
This has a natural symplectic form (the Fubini-Study form) we do not pause to write
down, and a Hamiltonian action of T n+1, whose Lie algebra we can identify with Rn+1.
One moment map is then


|z0|2
|z1|2
|zn|2
Φ([z0, z1, . . . , zn]) = P.
,P,... , P
2
2
2
|z
|
|z
|
i
i
i
i
i |zi|
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The individual coordinates can only vary between 0 and 1, and their sum is automatically1; it is easy to check that the image is the whole simplex. In fact this is really the λ =(1, 0, 0, . . . , 0) case of the Schur-Horn result.
2.3. The relation with the Schur-Horn theorem. Let K be the unitary group U(n), and T
the subgroup of diagonal matrices, a “maximal torus” of U(n). The Lie algebra u(n) is thespace of skew-Hermitian matrices.2 We can identify the Hermitian matrices with u(n)∗using the trace form, H 7→ Tr (iH·). This identification is U(n)-equivariant, intertwining
the conjugation action with the coadjoint action; in particular it takes orbits of Hermitianmatrices to coadjoint orbits.
The upshot is that we can think of coadjoint orbits of U(n) as orbits of U(n) acting on
the space of Hermitian matrices by conjugation, which of course are the isospectral setsOλ required for the Schur-Horn theorem.
From this point, we just have to apply properties 1 and 2 of moment maps; Oλ is a
symplectic manifold, and the action of the diagonal subgroup T is Hamiltonian with moment map Oλ ֒→ u(n)∗ →→ t∗ . Using our trace form identification, this composite is themap taking a Hermitian matrix with spectrum λ to its diagonal entries. Then the A/G-S
convexity theorem tells us that the image is the convex hull of images of T -fixed points. A
Hermitian matrix is fixed under conjugation by all diagonal matrices if and only if it itself
is diagonal, which means its entries are a permutation of λ. This proves Schur-Horn.
(Of course this presentation is historically unfair, as indicated at the beginning – theSchur-Horn theorem was a primary inspiration for A/G-S convexity.)
2.4. Stabilizer groups. We mention one of the many easy properties of the definition of
moment map. The stabilizer algebra of a point m for the action of a Lie algebra k isdefined as the Lie subalgebra giving vector fields that vanish at m. (Correspondingly,
the flows they generate leave m fixed, so the stabilizer algebra is the Lie algebra of thestabilizer group of m.)
Proposition. Let M be a Hamiltonian K-manifold, with moment map Φ, and m ∈ M. Then thestabilizer algebra of m is the perp of the image of the differential dΦ mapping from the tangent
space at m to k∗ .
(Again we’re using the natural pairing between k and k∗ ; the perpendicular of a subspace of k∗ is a subspace of k.)
Proof. A vector ~k is in the stabilizer algebra of m iff the vector field induced on M vanishes at m, iff the symplectic gradient of hΦ, ~ki vanishes at m, iff the differential dhΦ, ~kivanishes at m, iff hdΦ, ~ki vanishes at m, which means ~k is in the perp of the image of thedifferential dΦ at m.
There are two interesting extremes of this. One is that m’s stabilizer group in K is
discrete if and only if m’s stabilizer algebra in k is trivial if and only if the differential of Φ
at m is onto. On the other hand, m is a K-fixed point if and only if all of the vector fields
induced by k vanish at m if and only if the differential of Φ at m is zero.
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A unitary matrix has UU∗ = 1. A unitary matrix near the identity, 1+ǫS, therefore has (1+ǫS)(1+ǫS)∗ =1 + ǫ(S + S∗ ) + O(ǫ2 ) = 1. Differentiating, we get S = −S∗ .
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Corollary. Let p ∈ k∗ be a boundary point of the image of a moment map Φ : M → k∗ . Then eachpoint in Φ−1(p) is stabilized by at least a circle subgroup of K.
Proof. If p is an extremal point, the image of the differential can’t be onto – some directions
from p lead outside the image of Φ. So the perp to the differential is positive-dimensional,
and generates a positive-dimensional subgroup of K, the connected component of thestabilizer group. And any positive-dimensional compact group contains a circle group.

We include these proofs to show off the simple connection between the action of the Liegroup and the properties of a moment map. While A/G-S convexity is based on Morsetheory applied to the moment map, the results above are much more pedestrian and use
only basic differential geometry.
This doesn’t all come for free, of course – the important fact is that we happen to be
studying isospectral sets of Hermitians (which turn out to be symplectic manifolds), andcertain maps from them like “take diagonals” (which turn out to be moment maps). Most
other equations one might like to study in linear algebra are not statable as the vanishingof a moment map. I personally take this framework as a guide to some of those linear
algebra problems which are likely to have nice solutions.
2.5. When K is not a torus. There is a different, rather less pleasant, convexity result fornoncommutative groups K. The first example is K = SO(3) the group of rotations of R3,
and M a coadjoint orbit. We can identify so(3)∗ with R3 and so M is a sphere centeredat the origin. Since this is not convex, we know we’ll have to look for a slightly subtler
statement than A/G-S.
In the conjugation action of U(n) on the space of Hermitian matrices (not naturally a
symplectic manifold), we know two nice things already; every Hermitian matrix is diagonalizable, and if we insist that the (real) diagonal entries then be in decreasing order, thediagonal matrix is unique. Inside the space of diagonal Hermitian matrices, which we
can think of as the dual t∗ to the Lie algebra of the maximal torus of U(n), this picks out acertain cone t∗+ called a positive Weyl chamber for U(n). Since U(n) is the only example
we will need we won’t give the general definition of positive Weyl chamber, but merely
state that for every connected compact Lie group K, there is an analogous group T andpolyhedral cone t∗+, such that each orbit in k∗ intersects t∗+ in a unique element.
Theorem. Let M be a compact connected Hamiltonian K-manifold, with moment map Φ. Thenthe intersection of the image of Φ with the positive Weyl chamber t∗+ is a convex polytope.
This theorem is due to Kirwan, and is a fair bit harder than the case K commutative(the A/G-S theorem above). The new difficulties primarily come from the points of the
moment polytope lying on the boundary of the positive Weyl chamber (in the Hermitiancase: the Hermitian matrices with repeated eigenvalues), which we call the Weyl walls.
We can ignore these to some extent: define the symplectic slice of M as the preimage
under Φ of the interior of t∗+. It is then a theorem that the symplectic slice is symplectic,
and a Hamiltonian T -manifold, whose image is the K-moment polytope minus the parts
hitting the Weyl walls. (Since it is usually noncompact we can’t use it to trivially reduce
Kirwan convexity to A/G-S convexity.)
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There is a slightly different statement of Kirwan convexity: instead of intersecting theimage of Φ with t∗+, we can compose Φ with the K-invariant map k∗ → t∗+ that takes anelement to the unique point in t∗+ in its K-orbit. (For K = U(n) this takes a Hermitianmatrix M to the diagonal matrix with M’s eigenvalues decreasing down the diagonal.)
Then Kirwan convexity says that the image of this composite map is a convex polytope(the same as constructed before by intersecting).
With all this machinery we can give pleasant proofs of Horn’s theorems (but not Horn’sconjecture!) on the sum of two Hermitian matrices.
Theorem. Let Oλ, Oµ be the spaces of Hermitian matrices with spectrum λ, µ. Let e : Oλ×Oµ →Rn take a pair of matrices to the spectrum of their sum, listed in decreasing order. Then the imageof e is a convex polytope. Also, if e(Hλ, Hµ) is an extremal point of the image of e and is also astrictly decreasing list, then Hλ, Hµ are simultaneously block diagonalizable.
Proof. We have already explained how to identify Oλ and Oµ with coadjoint orbits ofU(n). Therefore by property 3 of moment maps listed above, their product has a Hamiltonian action of U(n) × U(n). (In fact it is a coadjoint orbit for this big group.)
Consider the action of the diagonal U(n), i.e. conjugating both Hermitian matrices
by the same unitary matrix. Then by property 1 this action is Hamiltonian, and we cancompute its moment map as the transpose of the inclusion u(n) ֒→ u(n) ⊕ u(n), composedwith the U(n) × U(n) moment map, which was just inclusion of the coadjoint orbit.
The transpose of diagonal inclusion V → V ⊕ V is summation V ∗ ⊕ V ∗ → V ∗ ; so themoment map for U(n)’s action on Oλ × Oµ just takes a pair of Hermitian matrices to theirsum.
But now we have convexity, as our map e is just the map used in the alternate description of Kirwan’s convexity theorem.
If e(Hλ, Hµ) is a strictly decreasing list of real numbers, that means it’s in the interiorof the positive Weyl chamber, therefore in the image of the symplectic slice, which for
Oλ ×Oµ is the set of pairs (Hλ, Hµ) whose sum is diagonal with decreasing entries (alreadya familiar set to people studying Horn’s problem). Then we can use some element ofU(n) to conjugate (Hλ, Hµ) into the symplectic slice, on which e is the moment map forthe action of T .
Since e(Hλ, Hµ) is on the boundary of the image of e, we can apply the corollary from
subsection 2.4, and determine that (Hλ, Hµ) is invariant under some circle in T (and notjust the scalar matrices, which fix all pairs (Hλ, Hµ)). Being invariant under conjugation
by a nonscalar diagonal matrix forces each of Hλ and Hµ to be block diagonal.
(The very careful reader will wonder why we bothered with the symplectic slice in theabove, since the proposition about stabilizers didn’t require that the group be a torus. But
just because we knew that e(Hλ, Hµ) was on the boundary of the image of e, we didn’tknow Hλ + Hµ to be on the boundary of the image of U(n)’s moment map – and in fact it
potentially wasn’t, if e(Hλ, Hµ) was on a wall of the Weyl chamber.)
There is now an industry generalizing these convexity results to larger contexts (suchas “Poisson” actions), some of which are definitely relevant to linear algebra; we don’tpause to discuss these, as we won’t need them for Horn’s problem.

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3. S YMPLECTIC QUOTIENTSGiven M a Hamiltonian K-manifold with moment map Φ : M → k∗ , we now havea number of theorems about the image of Φ – put differently, about which fibers of Φ
are nonempty – and the action of K on the fiber. Given this setup, and a point µ ∈ k∗
fixed by K, define the symplectic quotient (or symplectic reduction) of M at the levelµ as Φ−1(µ)/K. There is a reason that these quotients are nicer to study than the fibers
themselves:
Theorem (Marsden-Weinstein, 1974). The (dense, open) smooth part of Φ−1(µ)/K inherits acanonical symplectic structure.
(The case most frequently studied is µ is a regular value of Φ, since Φ−1(µ) is then asubmanifold by the inverse mapping theorem, and the action of K on it has only finite
stabilizers as discussed in subsection 2.4.)
In the case K = T , where the conjugation action is trivial, the adjoint and coadjoint
representations are also trivial. So any point µ ∈ t∗ is fixed by T . In the case K = U(n)only the scalar Hermitian matrices are fixed by conjugation.
Example 5. Let M = Cn, with the circle group S1 = {exp(iθ)} acting by multiplication byphases. Identifying the Lie algebra of S1 with the reals, the moment map is Φ(~v) = 21 |~v|2.
Then if k > 0, the symplectic quotient Φ−1(k)/S1 is CPn−1; this is the Hopf fibration, fibering S2n−1 by circles. If k = 0, the symplectic quotient is a point. If k < 0, the symplectic
quotient is empty.
Example 6. Let M = Oλ be the space of Hermitian matrices with spectrum λ, and K = T
the group of diagonal unitary matrices acting by conjugation, so Φ is the map taking a
Hermitian matrix to its diagonal entries. Then Φ−1(µ)/T is the space of Hermitian matrices with eigenvalues λ, and diagonal entries µ, up to conjugation by diagonal unitary matrices. These spaces are studied in the doctoral theses [Kn, G], and the caseλ = (1, 1, 0, . . . , 0) is described in detail in subsection 9.1.
4. F LAG MANIFOLDS
Hopefully the previous section has convinced the reader that some of the fundamental
objects of interest in this theory are the isospectral sets themselves. We will see now thatthese are not just real manifolds, but complex manifolds, suggesting that they may bestudiable using complex algebraic geometry.
Define a (partial) flag in Cn as an increasing list V of subspaces 0 = V0 < V1 < . . . <Vs = Cn, with relative dimensions dim Vi/Vi−1 = di. If s = n then V is called a full flag. If
s = 2 then V is just given by a subspace of dimension d1 (with the automatic V0 and V2).
Given a Hermitian matrix H, we can associate a partial flag VH as follows: let Vi be thesum of the eigenspaces corresponding to the i smallest eigenvalues. This flag will be full
if and only if H has no repeated eigenvalues. Conversely, given a partial flag V and anincreasing list of s eigenvalues ei, we can construct a Hermitian matrix HV as the sum
HV =

s
X

⊥
ei · [the projection onto Vi−1
∩ Vi].

i=1

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So in all, the space of Hermitian matrices with spectrum λ is in 1:1 correspondence with acertain space of partial flags we denote Flags(d1, . . . , ds) (where the {di} are determinedby λ’s repetition of eigenvalues), called a flag manifold. (In the special case s = 2, this isalso called the Grassmannian of d1-planes and denoted Grd1 (Cn).)
What have we gained? The benefit of this view is that while Hermitian matrices only
have an evident action of the unitary group U(n), the space of flags Flags(d1, . . . , ds) has
an action of the group GLn(C) of all invertible n × n matrices – applying a linear transformation to a flag produces a new flag. Since U(n) acts transitively on the Hermitianmatrices with a given set of eigenvalues, so too will this larger group GLn(C) act with one
orbit – given two flags, there is a linear transformation taking one to the other. (And it
may even be taken to be unitary.)
The unitary matrices stabilizing a given Hermitian matrix – say, a diagonal one withstrictly decreasing eigenvalues – are easy to understand; they are just the diagonal unitarymatrices T . From this we can conclude that if λ has no repeated eigenvalues, we can identify the space of Hermitian matrices with spectrum λ with U(n)/T . For the corresponding
statement for GLn(C) we must compute the stabilizer of a flag – say, the standard flag
(0 < C1 < C2 < . . . < Cn) – which can be seen to be the group B of invertible upper
triangular matrices. So we can identify Flags(1, 1, . . . , 1) with GLn(C)/B.
Since GLn(C) is not just a real Lie group but a complex Lie group, and B a complex
subgroup,3 we find out that Flags(1, 1, . . . , 1) is naturally a complex manifold. As wewill see in the sections to come, it is actually a “complex algebraic variety”.
We invite the reader to determine the corresponding statements for the s < n, partial
flag manifold case.
5. G EOMETRIC INVARIANT THEORY IN THE AFFINE CASE
Our interim goal is to describe the algebro-geometric analogue of symplectic quotients.
We first give the basics of affine algebraic geometry, and the concept of quotient in thatcase; however for our linear algebra applications we’ll need to work through the additional complications of projective geometry.
Before getting into algebraic geometry, let us think about embeddings of manifolds
M into Rn. On the one hand, if we choose n real-valued functions {fi} on M, suchthat any two points are distinguished by at least one of the functions, the map m 7→
(f1(m), f2(m), . . . , fn(m)) gives an injection4 of M into Rn. Conversely, if M is a submanifold of Rn, then the n coordinate functions, restricted to M, give us n functions separating
points.
By taking polynomials in those n functions, we get an algebra of functions on M, somequotient of the polynomial ring R[f1, . . . , fn]. For example, if M is the parabola y = x2
in R2, and f1 = x and f2 = y, then the algebra of functions they will generate will be
R[f1, f2]/(f21 − f2). (The function f21 − f2 is not zero on the whole plane, but it is the zerofunction on M.)
3This means that its Lie algebra is invariant under multiplication by i. By contrast, U(n) is not a complex
subgroup because i times a skew-Hermitian matrix is not always skew-Hermitian.
4
Which is not good enough, if we’re thinking about smooth manifolds M and want to model the smooth
structure; we must also ask that the map be an immersion. This goes by the phrase “separating points”,
which we’re already asking, and “separating tangent vectors.”
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The manifolds we are actually interested in are flag manifolds (and things we buildfrom them), which we saw in the last section are complex. For this reason the only rings
we will bother to consider henceforth will be quotients of polynomial rings with complex
coefficients rather than the reals.
5.1. The spectrum of a ring. Let’s now think about the reverse direction: given a quotient
R = C[f1, . . . , fn]/I of polynomial ring by an ideal, we can define the spectrum5 Spec R
as the subset of Cn where all the polynomials in the ideal I vanish.
Of course, this is a bad definition, as the notation Spec R only refers to the abstract ringR, and not the particular way of presenting it as a quotient of a polynomial ring. (Theremay be many of these, just as there are many embeddings of M into affine spaces, of many
dimensions.) There is an alternate, equivalent, definition of Spec R as the set of maximal
ideals6 of R, which doesn’t require us to choose an embedding. But just as we generally
deal with a manifold by picking coordinates on it, we generally deal with affine varieties
Spec R by embedding them in affine space.
Given a map on rings R → S, there is an induced map7 from Spec S to Spec R, taking amaximal ideal to its preimage. The most fundamental example of this is C[x1, . . . , xn] →→ R,
which induces the inclusion Spec R ֒→ Cn. Not every function from Spec S to Spec R arisesin this way, only those preserving in some sense the structure of algebraic variety. While
it is possible to give a definition of such functions without explicit reference to R and S
we will not need this.
5.2. Group actions. Now take the situation that a complex Lie group G acts on our ring R
by ring automorphisms. For example, if R = C[x1, . . . , xn] (so Spec R is just Cn itself), then
GLn(C) acts on R by linear changes of the {xi}, and the induced action on higher-orderpolynomials.
Since G acts by ring automorphisms, it takes maximal ideals to maximal ideals, so actson the set Spec R. We will be interested in forming the quotient (Spec R)/G, with thenatural map Spec R →→(Spec R)/G.

Since we’re thinking in terms of algebraic geometry, this really means we’re looking
for a ring S with Spec S = (Spec R)/G; equivalently, S should be the ring of functions on(Spec R)/G. Each element of S therefore pulls back to a G-invariant function on Spec R.
So S will have to map back to the G-invariant functions on Spec R, which we denote RG;
this is the clear candidate for S. Since RG is the “ring of invariants”, we will call Spec (RG)
the geometric invariant theory quotient (or GIT quotient) of Spec R by G, and denote it(Spec R)//G.
It is not quite true, though, that (Spec R)//G = (Spec R)/G; rather we only have a map(Spec R)/G → (Spec R)//G. It is already a tricky theorem that when G is “reductive” (likeGLn(C), and all the other groups we will be using) that (1) RG is finitely generated, so
5

If T is a linear transformation V → V, and R = C[T ]/( the characteristic polynomial of T ), then Spec R is
the spectrum in the usual sense. It is rather amusing to trace in this way the modern-day algebraic geometry
terminology back to Rydberg lines of atoms!
6
One of the great insights of algebraic geometry in this century is that for almost all purposes one should
instead work with the prime ideals, a somewhat larger set than the maximal ideals. But we will be avoiding
any of the contexts in which this distinction becomes important.
7
For the rings we will consider, which are finitely generated; geometrically, this corresponds to us being
able to embed our varieties in finite-dimensional vector spaces
10(Spec R)//G can be embedded in a finite dimensional space, and (2) the natural map from(Spec R)/G → (Spec R)//G is onto.

Example 7. Let G = C× , the nonzero complex numbers under multiplication, acting on
R = C[x1, . . . , xn] by rescaling each coordinate the same way. Then the induced action
on G on Cn is also by rescaling. The ordinary set-theoretic quotient is not Hausdorff; it’s
projective space union the point ~0 in the closure of every other point. The only invariant
polynomials in RG are the constants, so Spec (RG) = pt. (This will turn out to be related to
the k = 0 case of example 5.)
Obviously this is unsatisfying; we’d rather the quotient be projective space itself (somehow including the k > 0 case of example 5). But projective space is not Spec S for any S, by
Liouville’s theorem – any function on projective space is constant! We will need subtler
constructions than Spec to make projective spaces, and more to the point, flag manifolds.
6. G EOMETRIC INVARIANT THEORY IN THE PROJECTIVE CASE , AND THE K IRWAN /N ESSTHEOREMLet R = ⊕k∈N Rk now be a graded ring, meaning that the product of an element of Rk
with an element of Rm lands in Rk+m. The standard example is R a polynomial ring, with
Rk the homogeneous polynomials of degree k.
We will define Proj R in several equivalent ways. The simplest, least useful for visualizing examples, is that Proj R the set of maximal graded ideals of R, where a graded ideal
is one equal to the direct sum of its intersections with the Rn.
For the second, note that R has a natural action of C× , acting on Rk by rotating it withspeed k. There is a natural C× -invariant surjection R →→ R0 (which would not be true if R
were Z-graded instead of N-graded). So there is a map backwards Spec R0 ֒→ Spec R. Then
Proj R := (Spec R r Spec R0)/C× .

Example 8. Let R = C[x1, . . . , xn] with Rn the homogeneous polynomials of degree n.
Then as discussed in example 7, R0 is just the constants, and the point Spec R0 includes
into Spec R = Cn as the origin ~0. Then Proj R = (Cn r {~0})/C× is just the usual definitionof CPn−1.
So Proj of a polynomial ring (with each variable given degree 1) is projective space,just like Spec of a polynomial ring is affine space. And just as writing a ring R as the quotient of a polynomial ring, C[x1, . . . , xn] →→ R, dually gives us an inclusion of Spec R ֒→ Cn,
writing a graded ring R as the quotient of a polynomial ring by a graded ideal I dually
gives us an inclusion Proj R ֒→ CPn−1. This gives a third description of Proj R, in the case
R is presented as C[x1, . . . , xn]/I and all the xi are degree 1; as the points in CPn−1 where
the elements of I all vanish. In this case Proj R is called a projective variety.
The description we will use most is the second, in terms of Spec R and Spec R0.
6.1. Another relation between Spec and Proj . Given an ungraded ring R0, we can define
R := R0[l], where l is a new variable assigned formal degree 1. Then Spec R = Spec R0 × C,
so Spec R r Spec R0 = Spec R0 × C× , and Proj R = Spec R0. Upshot: anything we can makeas a Spec , we can also make as a Proj .
11

6.2. Maps of graded rings. Given a ring homomorphism R → S, we defined a natural
map from Spec S → Spec R. So it is natural to assume that a grading-preserving map
R → S induces a map on the corresponding Proj ’s. This is not the case! Rather:
Proposition 1. Let f : R → S be a homomorphism of graded rings. Let X be the points ofSpec S r Spec S0 such that every function in f(R) vanishes at X, and define (Proj S)usf as the×quotient of X by the natural action of C . Then f does not necessarily induce a map from Proj S →Proj R, but does induce one from Proj S r (Proj S)us
f → Proj R.

Proof. To see what the problem is, let’s trace through the definition of Proj . Since R and S
are graded, i.e. have C× -actions, Spec R and Spec S have C× actions. The map between therings being grading-preserving is equivalent to it being C× -equivariant. Consequently,
the map backwards on Spec S → Spec R is C× -equivariant.

The problem comes when we try to rip out Spec S0 and Spec R0. Of course there is no
problem when in restricting the map to Spec S r Spec S0. But the image of this may notland inside Spec R r Spec R0; there may be points of Spec S r Spec S0 that hit Spec R0. Once
we rip them out, then there’s no problem.
In the context to come, the set (Proj S)usf will be called the unstable set (hence the us).
6.3. Geometric invariant theory. We now apply what we’ve learned about maps of Proj ’s
to the case we studied in the last section, RG ֒→ R. Given a graded ring R with an action ofa group G, define the geometric invariant theory quotient(Proj R)//G := Proj (RG).
There is a natural map from Proj R minus the unstable set described by the proposition
above (the set of points where all G-invariant functions vanish) to (Proj R)//G.
While this may seem like a totally canonical definition, there are two traps for the unwary. In many problems one starts with the variety Proj R and its G-action rather than thering R, and does not want the extra choice that comes in finding an R (of which, unlike inthe affine case, there will be many). And even when one has chosen an R, there may be
many ways to get G to act on it inducing the same action on Proj R. It turns out that boththese choices matter – which is to say, the notation (Proj R)//G is misleading.
Example 9. Let X = Cn, and G = C× act by rescaling. Then X = Spec C[x1, . . . , xn] =Proj C[x1, . . . xn, l], where the {xi} are all degree 0, and l is degree 1. The action of G on
our homogeneous coordinate ring R is not quite determined – while the xi are required toall be weight 1, the action on the variable l can be any weight k, i.e. z · l = zkl.
If k > 0, there are no invariant polynomials other than constants: RG = C, and Proj (RG)
is the empty set.
If k = 0, the invariant ring RG = C[l], and Proj (RG) is a point.
If k = −1, the invariant ring RG = C[x1l, x2l, . . . , xnl] with all products of degree 1,
so Proj (RG) = CPn−1. It is a little trickier to see that this is true for all negative k. (It isinteresting to compare this to example 5. The difference in signs can be explained but we
do not do so here.)
Example 10. Let X = Cm×n, the space of m × n matrices, with m ≤ n, and let G = GLm(C)act on X on by left multiplication. Then as in the previous example (the m = 1 case of this12one) there is a parameter k describing the action of G on the extra variable l, such that for
k > 0 the quotient X//G is empty, and for k = 0 the quotient is a point.
Regard X as the space of m vectors in Cn. For k < 0, the unstable set turns out to be theset of linearly dependent m-tuples, and the GIT quotient X//G is thus the Grassmannian
Grm(Cn). (If you take a linearly independent set of m vectors, and quotient by the actionof GLm(C), you forget the vectors and only remember the m-subspace they span.)
6.4. The equivalence of symplectic and GIT quotients. We spell out a moment map
that’s a special case of things already said: complex projective space CPn−1 is a Hamiltonian U(n)-manifold, with moment map
CPn−1 → u(n)∗[~v] 7→ [the rank 1 projection onto C~v]

using the identification already given between u(n)∗ and the space of Hermitian matrices.
In fact we are observing here that CPn−1 is a coadjoint orbit of U(n), the orbit O(1,0,...,0).
It is an easy fact,8 not proven here, that any complex submanifold of CPn−1 inherits a
symplectic structure from CPn−1. If X ⊆ CPn−1 is preserved by the action of a subgroup
K ≤ U(n), we can compute the moment map using the properties we listed of momentmaps:
X ֒→ CPn−1 → u(n)∗ → k∗ .

Lastly, given an action of a compact group K on a complex vector space, there is aunique extension to an action of the complexification KC of K. We will not stop to define this group in detail; suffice it to say that every compact group is a subgroup of aunique complex group, such that the Lie algebra of the complex group is the complexification of the Lie algebra of the compact group. The only example that will interest us
is U(n)C = GLn(C) (every matrix is uniquely the sum of a skew-Hermitian and i times askew-Hermitian).
We are ready to state the deepest theorem in this paper, proven separately by Kirwan
and Ness:
Theorem 1. Let K act on the graded ring R = C[x1, . . . , xn]/I, all generators xi of degree 1. So
Proj R is a subvariety of CPn−1, and (if smooth) a Hamiltonian K-manifold. Let Φ be the momentmap calculated above for K’s action on M. Then there is a natural identification∼ (Proj R)//KC
Φ−1(0)/K =between the symplectic quotient and geometric invariant theory quotient.
In fact there is a more general result, slightly more complicated to state, in which Proj R
is not necessarily projective. We already saw this in the case of K = S1 acting on Cn, in
examples 5 and 9.
7. B ACK TO FLAG MANIFOLDS
Since we’ve only stated the symplectic quotient equals GIT quotient theorem at thelevel Φ = 0, we need to slightly modify the Horn problem A + B = C to A + B + C = 0.
8

once one proves the non-easy Darboux theorem, given the unusual way we’ve defined symplectic
13Let M = Oλ × Oµ × Oν, the space of triples of Hermitian matrices with eigenvalues
respectively λ, µ, ν. Then the moment map for the diagonal action of U(n) on the productis the sum of the three matrices, and Horn’s problem asks when this symplectic quotient
is nonempty.
To jack into the symplectic vs. GIT theorem, we need to express Oλ as Proj of something, or equivalently explain how to embed it in projective space such that the restrictionof the symplectic structure on projective space matches that on Oλ. This turns out to only
be possible if λ is integral, and involves some very classical geometry.
7.1. The U(n) case of the Borel-Weil-Bott-Kostant theorem. In fact we will not need thedetails presented in this section in what follows, but it lets us avoid some representation
theory of GLn(C).
Recall the flag manifolds Flags(d1, . . . , ds) defined before; we will restrict to the casedi ≡ 1 and just write Flags(Cn), leaving the interested reader to work out the generalcase. Denote by Grk(Cn) the Grassmannian of k-dimensional subspaces of Cn (these arevery partial flag manifolds). There is a natural forgetful map from Flags(Cn) to each
Grassmannian, taking a flag to its subspace of dimension k, forgetting all the subspaces
below and above.
Since a flag is just a list of subspaces, the product mapnFlags(C ) →nY

Grk(Cn)k=1

is an inclusion.
Given a k-dimensional subspace A of Cn, we can wedge together a basis {~ai} of A tok n
~ i only
get a nonzero alternating tensor in ∧ C . If we change the basis, our element ∧ki=1achanges by a scalar factor, so gives a well-defined element of the projective space. This iscalled the Plücker embedding
Grk(Cn) → P(∧kCn).

Given a vector ~v ∈ V, and a natural number a, we can tensor ~v with itself a times
to get a symmetric tensor ~v⊗a ∈ Sym a(V). This descends to the projectivized spaces
PV ֒→ P(Sym aV), and is called the ath Veronese embedding. Choosing a family {ak} ofnaturals, we get maps
P(∧kCn) → P(Sym ak (∧kCn)).

In the very special case that dim V = 1, we don’t need a to be a natural number – for a < 0
we define Sym a(V) := Sym −a(V ∗ ). This will be handy in the case k = n, where ∧nCn is
one-dimensional.
~ ∈ W, we can tensor them together, inducing a map
Finally, given vectors ~v ∈ V and w
on projective spaces PV × PW → P(V⊗W) called the Segre embedding. In the case athand, this gives a map
n
Y

k=1

ak
P(Sym ak (∧kCn)) → P(⊗n(∧kCn)).
k=1Sym14

Proposition (one small aspect of the Borel-Weil-Bott-Kostant theorem). Let λ = (λ1 > . . . >λn) be a strictly decreasing list of integers, and ak = λk − λk+1 for k = 1 . . . n (taking λn+1 to bezero). Then the composite of the above maps
akFlags(Cn) → P(⊗n(∧kCn))
k=1Syminduces a symplectic structure on Flags(Cn) matching the one given by its diffeomorphism withthe coadjoint orbit Oλ.
(Again, we exhort the reader to think about the case of λ only weakly decreasing, thepartial flag manifold case.)
Since we have exhibited Oλ as a variety embedded in projective space, we can look at its
homogeneous coordinate ring, which we will denote Rλ, reserving subscripts to indicate
graded pieces.
7.2. The Borel-Weil theorem. It is evident that all the maps described above are equivariant with respect to the action of GLn(C) on Cn, so GLn(C) therefore acts on the homogeneous coordinate rings Rλ, and in particular each graded piece (Rλ)k is a representationof GLn(C). We pause to mention three important facts about these representations.
Proposition 2 (part of Borel-Weil). 1. Each graded piece (Rλ)k of each homogeneous coordinate ring for the flag manifold is an irreducible representation of GLn(C).
2. (Rλ)k is isomorphic, as a representation, to (Rkλ)1.
3. Every irreducible representation of GLn(C) arises as (Rλ)1 for a unique λ.
We denote (Rλ)1 by Vλ; for reasons we will not go into here it is called the irreducible
representation of GLn(C) of highest weight λ. So we can identify Rλ with ⊕n∈N Vnλ.
A little bit more can be said, for those who know the classification of irreducible representations of Lie groups in terms of “highest weights”, but as representation theory is notin the title of this paper we will not explore this further.
8. E ASY CONSEQUENCES
We need one more fact about the homogeneous coordinate rings Rλ for the flag manifold: they have no zero divisors. This is implied, though we will not explore how, by the(much stronger) fact that the flag manifold is connected and smooth.
We’re now ready to apply our big gun, the equivalence of symplectic and GIT quotients.
Theorem. Let λ, µ, ν be weakly decreasing sequences of integers. Then the space(Hλ, Hµ, Hν) : Hλ + Hµ + Hν = 0
U(n)can be identified withProj

M
(Vkλ⊗Vkµ⊗Vkν)GLn (C).
k∈N

Proof. We’ve already seen that the first space is the symplectic quotient of Oλ ×Oµ ×Oν by
the diagonal action of U(n), at the level 0. By the symplectic vs. GIT equivalence, this is
the geometric invariant theory quotient of a certain product of flag manifolds by GLn(C).
15The Borel-Weil-Bott-Kostant theorem, or rather the small part of it presented here, explains how to embed Oλ, Oµ, Oν into projective space in order to be able to apply the
symplectic vs. GIT theorem. The individual coordinate rings are ⊕n∈N Vnλ (likewise µ, ν);the coordinate ring of the product is, by Segre embedding, ⊕n∈N (Vnλ⊗Vnµ⊗Vnν)GLn (C).
Then to take the GIT quotient we take Proj of the invariant subring.
This explains a couple of connections already noticed in the literature between the
Horn’s problem and the representation theory of GLn(C):Corollary. Let Vλ, Vµ, Vν be three irreducible representations of GLn(C). If there is a GLn(C)invariant vector in the triple tensor product, then there exist Hermitian matrices Hλ, Hµ, Hν withthe corresponding spectra and zero sum.
Proof. If the tensor product has an invariant vector, then the coordinate ring of the GIT
quotient is nontrivial in its degree 1 piece. Since this ring is a subring of a ring with nozero divisors, it itself has no zero divisors, and therefore is nontrivial in all degrees. Thisis enough to conclude that Proj of it, the GIT quotient, is nonempty.
By the identification in the theorem, the symplectic quotient is therefore nonempty. So
there are three Hermitian matrices with the desired spectra adding to zero.
Corollary. Let λ, µ, ν be weakly decreasing lists of integers. If there exist Hermitian matrices
Hλ, Hµ, Hν with the corresponding spectra and zero sum, then for some k > 0, the tensor product
Vkλ⊗Vkµ⊗Vkν has a GLn(C)-invariant vector.
Proof. The condition says the symplectic quotient is nonempty, so running the theorem inreverse, the GIT quotient is nonempty. Consequently the ring of invariants is nontrivial.
Therefore in some graded piece n > 0 it is nontrivial, giving the desired result.
(Both of these corollaries appear in Klyachko’s work; he essentially repeats the proof of
Kirwan/Ness in this special case.)
Obviously there is a mismatch here; we would like to know that the Hermitian problem has a solution if and only if the tensor product has an invariant vector. But the big
machine presented here is not powerful enough to rid us of this k. Other techniques arenecessary and the first proof that k can indeed be taken to be 1 appeared in [KT].
9. C ONCLUSIONS , AND AN AMUSING EXAMPLE
We saw that certain famous equations in linear algebra, such as Hλ + Hµ = Hν (orbetter, Hλ + Hµ + H−ν = 0), are the conditions that a certain moment map be equal to zero.
Whenever this happens we can plug into the theory of Hamiltonian actions on symplectic
manifolds. The most spectacular results here establish the convexity properties of theimages of these moment maps, and go some way toward determining the image.
In fact the details of the proofs are not actually very different from the hands-on techniques used e.g. by Horn himself; the benefit here is in establishing a framework thatpoints out which problems are likely to accede to such local analysis, and in particularwhich are likely to lead to linear inequalities and polytopes.
In the (frequent) case that the symplectic manifolds under study are algebraic varieties,
one then has a totally new viewpoint, replacing the spaces by homogeneous coordinate
16

rings, and the study of the moment map by invariant theory. This connects up the symplectic problem with representation theory problems, illuminating both.
Lastly, we emphasize another lesson (not really applied in this paper) from both symplectic and algebraic geometry; the questions like “for which λ, µ, ν is the following symplectic quotient nonempty” should really be viewed as just the first step in a more detailed study of the symplectic quotient itself. What is its dimension? symplectic volume?
its Betti numbers, and cohomology ring?
9.1. An amusing example. Let X = C2×n, the space of 2 × n matrices, with its left actionof U(2) and right action of U(n). These actions are Hamiltonian with moment maps
M 7→ −MM∗ and M 7→ M∗ M. We will only be interested in the action of the diagonal
subgroup T n ≤ U(n), whose moment map picks out the diagonal entries of M∗ M. Notethat Tr MM∗ = Tr M∗ M, so the two moment maps are not entirely unrelated.
We will symplectic quotient by both groups U(2) and T n. The symplectic quotient by
U(2) at level s1 asks that our 2 vectors in Cn be orthogonal with norm-square s. So fors > 0, the symplectic quotient is the Grassmannian of 2-planes in Cn. (Which is good,
because we got this answer when we did this earlier by GIT, in example 10.) Since that isa coadjoint orbit of U(n), studying its reductions by T n is really the Schur-Horn problem
we first mentioned.
If we consider this in the opposite order, we get a very different picture. In symplecticquotienting C2×n by T n, each diagonal entry in T n acts on its own copy of C2. The symplectic quotient of C2 by U(1) is either empty, a point, or the Riemann sphere CP1, de∼ Rn. Let’s take each ai > 0 so the~ = (a1, . . . , an) chosen in tn =pending on the level set a1nsymplectic quotient by T is a product of CP ’s. In fact the moment map for the residual
action of U(2) on each of these CP1 identifies it with a sphere S2ai of radius ai.
Q
2
Think of such a sphere as the set of steps in R3 of length ai; then the product n
i=1 Saican be thought of as the set of n-step polygonal paths in R3, with ith step length ai.
Since the moment map for the diagonal action of U(2) on this product is the sum of theindividual moment maps, we can think of it as the function taking a path to its endpoint.
The symplectic quotient at level zero is then the set of polygons in R3 (because the moment map condition requires that the path terminate at the origin), with edge lengths {ai},considered up to rotation. The connection of polygon spaces to the Schur-Horn problem
was first noted in [HK].

R EFERENCES[F] W. Fulton, Young tableaux, with applications to representation theory and geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997.
[G] R. Goldin, The cohomology rings of weight varieties. Doctoral thesis, MIT 1999.
[GLS] V. Guillemin, E. Lerman, S. Sternberg, Symplectic fibrations and multiplicity diagrams. Cambridge
University Press, Cambridge, 1996.
[HK] J.-C. Hausmann, A. Knutson, Polygon spaces and Grassmannians. Enseign. Math. (2) 43 (1997), no.
1-2, 173–198.
[Kn] A. Knutson, Weight varieties. Doctoral thesis, MIT 1996.
[KT] A. Knutson, T. Tao, The honeycomb model of GLn (C) tensor products I: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), pp. 1055-1090.
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[MFK] D. Mumford, J. Fogarty, F. Kirwan. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer-Verlag,
Berlin, 1994.
E-mail address: allenk@math.berkeley.edu

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