A short geometric proof of a conjecture of Fulton

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by  Nicolas Ressayre

A short geometric proof of a conjecture of Fulton
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2 Abstract

We give a new geometric proof of a conjecture of Fulton on the
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G Littlewood-Richardsoncoefficients. This conjecture was firstly proved
A by Knutson, Tao and Woodwardusing the Honeycombtheory. A geo-
metricproofwasgivenbyBelkale. Ourproofisbasedonthegeometry
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of Horn’s cones.
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1 Introduction
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1 Recall that irreducible representations of Gl (C) are indexed by sequences
v r
3 λ = (λ1 ≥ ··· ≥ λr) ∈ Zr. If λr ≥ 0, λ is called a partition. Denote
3 the representation corresponding to λ by V . Define Littlewood-Richardson
λ
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coefficients cν ∈ N by: V ⊗V = cν V . W. Fulton conjectured that
3 λµ λ µ ν λµ ν
. for any positive integer N, P
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9 cνλµ = 1 ⇒ cNNνλNµ = 1.
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: ThisconjecturewasfirstlyprovedbyKnutson,TaoandWoodward[KTW04]
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i using the Honeycomb theory. A geometric proof was given by Belkale in
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[Bel07]. Theaim of this note is to give a shortproofof this conjecture based
r
a on the geometry of Horn cones.
NotethattheconverseofFulton’sconjectureisaconsequenceofZelevin-
ski’ssaturationconjecture. Thislastconjecturewasprovedin[KT99,Bel06,
DW00].
The key observations of our proof are:
(i) the non-trivial faces of codimension one of Horn cones corresponds to
Littlewodd-Richardson coefficients equal to one;
(ii) eachnonzeroLittlewood-Richardsoncoefficientgivealinearinequality
satisfied by Horn cones.
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Assume that cν = 1. By Borel-Weyl’s theorem, cν is the dimension of the
λµ λµ
Gl invariant sections of a line bundle L on a certain projective variety X.
r
Then, cNν is the dimension of the Gl invariant sections of L⊗N. This
NλNµ r
implies that cNν ≥ 1. In particular, cNν gives a linear inequality for
NλNµ NλNµ
a certain Horn cone: we prove that this inequality correspond to a face of
codimension one.
In this proof, the Littlewood-Richardson coefficients are mainly the co-
efficient structure of the cohomology of the Grassmannians in the Schubert
basis. Our technique can be applied to prove similar results for the coef-
ficient structure of the Belkale-Kumar’s product ⊙ on the cohomology of
0
others projective homogeneous spaces G/P.
2 Geometry of Horn cones
2.1 Horn’s cone of Eigenvalues
2.1.1— Schubert Calculus. Let Gr(a,b) be the Grassmann variety of
a-dimensional subspaces L of a fixed a+b-dimensional vector space V. We
fix a complete flag F : {0} = F ⊂ F ⊂ F ⊂ ··· ⊂ F = V. For any
• 0 1 2 a+b
subset I = {i < ··· < i } of cardinal a in {1,···,a+b}, there is a Schubert
1 a
variety Ω (F ) in Gr(a,b) defined by
I •
Ω (F ) = {L ∈Gr(a,b) : dim(L∩F ) ≥ j for 1 ≤ j ≤ n}.
I • ij
The Poincar´e dual of the homology class of Ω (F ) does not depend on F ;
I • •
it is denoted σ . The σ form a Z-basis for the cohomology ring. It follows
I I
that for any subsets I, J of cardinal a in {1,···,a+b}, there is a unique
expression
σ .σ = cK σ ,
I J IJ K
XK
for integers cK . We define K∨ by i ∈ K∨ if and only if a+b+1−i ∈ K.
IJ
Then, if the sum of the codimensions of Ω (F ), Ω (F ) and Ω (F ) equals
I • J • I •
the dimension of Gr(a,b), we have
σ .σ .σ = cK∨[pt].
I J K IJ
2.1.2— Horn’s cone. Let H(n) denote the set of n by n Hermitian
matrix. For A ∈ H(n), wedenoteits spectrumbyα(A) = (α ,···,α ) ∈Rn
1 n
repeated according to multiplicity and ordered such that α ≥ ··· ≥ α . We
1 n
set
∆(n) := {(α(A),α(B),α(C)) ∈ R3n : A, B, C ∈ H(n) s.t. A+B+C = 0}.
2
Set E(n) = R3n, let E(n)+ denote the set of (α ,β ,γ ) ∈ E(n) such that
i i i
α ≥ α , β ≥ β and γ ≥ γ for all i = 1,···,n −1. Let E(n)++
i i+1 i i+1 i i+1
denote the interior of E(n)+. Let E (n) denote the hyperplane of points
0
(α ,β ,γ ) ∈ E(n) such that α + β + γ = 0. The set ∆(n) is a
i i i i i i
closed convex cone containedPin E0(nP) and Pof non empty interior in this
hyperplane.
2.1.3—GIT-coneLetV beacomplexn-dimensionalvectorspace. Let
Fl(V) denote the variety of complete flags of V. The group G = Gl(V) acts
diagonalyonthevarietyX =Fl(V)3. LetusfixabasisinV,F ∈ Fl(V)the
•
standard flag for this base, B its stabilizer in G and T the torus consisting
of diagonal matrices. We identify the character groups X(T) and X(B)
with Zn in canonical way. The line C endowed by the action of B3 given
by (λ, µ, ν) ∈ (Zn)3 = X(B3) is denoted by C . The fiber product
(λ,µ,ν)
G3 ×B3 C(λ,µ,ν) is a G3-linearized line bundle Lλ,µ,ν on X; we denote by
L the G-linearized line bundle obtained by restricting the G3-action to
λ,µ,ν
the diagonal.
We denote by CG(X) the rational cone generated by triples of partitions
(λ, µ, ν) such that L has non zero G-invariant sections. The fist proof
λ,µ,ν
of the following is due to Heckman [Hec82], (see also [Ful00].
Theorem 1 Thecone∆(n)istheclosureoftherational convexconeCG(X).
2.2 Faces of ∆(n)
2.2.1— We have a complete description of the linear forms on E(n) which
define faces of codimension one of ∆(n). The first proof using Honeycombs
is due to Knutson,Tao and Woodward (see [KTW04]). A geometric proof is
due to Belkale ([Bel03]). In [Res07], I made a different geometric proof. A
proof using quivers is also given in [DW06].
Theorem 2 The hyperplanes α = α , β = β and γ = γ spanned
i i+1 i i+1 i i+1
by the codimension one faces of E(n)+ intersects ∆(n) along faces of codi-
mension one.
For any subsets I,J and K of {1,···,n} of the same cardinality such
that cK∨ = 1, the hyperplane α + β + γ = 0. intersects
IJ i∈I i j∈J j k∈K k
∆(n) along a face FIJK of coPdimensionPone. AnyPface of codimension one
intersecting E(n)++ is obtain is this way.
It is well known that if cK∨ 6= 0, for all (α,β,γ) ∈ ∆(n), we have
IJ
α + β + γ ≤ 0. In particular, the interstion betwenn ∆(n)
I i J j K k
P P P
3
and α + β + γ = 0 is a face F of ∆(n).
I i J j K k IJK
2P.2.2—WPenowrePviewsomenotionsof[Res07,Res08]andusenotation
of Paragraph 2.1.3. Let I,J and K be three subsets of {1,···,n} of the
same cardinality r such that cK∨ 6= 0. We associate to this situation a pair
IJ
(C,λ) where λ is a one parameter subgroup of G, and C is an irreducible
component of the set of fix points of λ in X. Consider the set C+ of the
x ∈ X such that lim λ(t)x ∈ C, and the parabolic subgroup P(λ) of G
t→0
associated to λ. The assumption cK∨ 6= 0 implies that the morphism
IJ
η : G× C+ −→ X,[g :x] 7−→ g.x,
IJK P(λ)
is dominant with finite general fibers. Now, F correspond to a face
IJK
FQ of CG(X): the entire points in FQ correspond to the line bundles
IJK IJK
L in CG(X) such that λ act trivialy on L . By [Res07, ] or [Res08, ], the
|C
Q
entire points in F correspond to the G-linearized line bundles L on X
IJK
such that Xss(L) intersects C.
Byconstruction,λactswithtwoweightsonV,thefirstonehasmultiplic-
ityrandtheotheronen−r. Inparticular,thecentralizerGλ inGofλisiso-
morphicto Gl ×Gl . Moreover, C is isomorphicto Fl(Cr)3×Fl(Cn−r)3.
r n−r
Now, consider the restriction morphism
ρQ : PicG3(X) −→ Pic(Gλ)3(C).
IJK
2.2.3— Let I,J and K be three subsets of {1,···,n} of the same
cardinality r. Define the linear isomorphism ρ by:
IJK
E(n) −→ E(r)⊕E(n−r)
(α ,β ,γ ) 7−→ ((α ) ,(β ) ,(γ ) )+((α ) ,(β ) ,(γ ) ).
i i i i i∈I i i∈J i i∈K i i∈/I i i∈/J i i∈/K
One easily checks that with evident identifications, ρ is obtained from
IJK
Q
ρ by extending the scalar to the real numbers.
IJK
Proposition 1 Let I,J and K be as above with cK∨ 6= 0. Let (α,β,γ) ∈
IJ
E(n)+. Then, (α,β,γ) ∈ F if and only if ρ (α,β,γ) ∈ ∆(r)×∆(n−
IJK IJK
r).
Proof. Assume that ρ (α,β,γ) ∈ ∆(r)×∆(n−r). Let A′,B′,C′ ∈ H(r)
IJK
and A′′,B′′,C′′ ∈ H(n−r)such that A′+B′+C′ = 0 and A′′+B′′+C′′ = 0
whose spectrums correspond to ρ (α,β,γ). Consider the three following
IJK
matrices of H(n)
A′ 0 B′ 0 C′ 0
A = , B = , C = .
(cid:18) 0 A′′ (cid:19) (cid:18) 0 B′′ (cid:19) (cid:18) 0 C′′ (cid:19)
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By construction, α is the spectrum of A and α = tr(A′), and similarly
I i
for B and C. We deduce that (α,β,γ) ∈ FIJKP.
By Theorem 1, we can prove the converse for the cone CG(X). Let
L ∈ F . Since Xss(L) intersects C, C contains semistable points for the
IJK
action of Gλ and ρQ (L). It follows that ρQ (L) ∈ ∆(r)×∆(n−r). (cid:3)
IJK IJK
Corollary 1 Let I,J and K be as in the proposition. Then, if F inter-
IJK
sects E(n)++, it has codimension one. In particular, cK∨ = 1.
IJ
Proof. By Proposition 1, F ∩E(n)++ is isomorphic to an open subset
IJK
of ∆(r)×∆(n−r). So, F has codimension 2inE(n) andsocodimension
IJK
one in ∆(n). Now, Theorem 2 implies that cK∨ = 1. (cid:3)
IJ
Remark. Corollary 1 for CG(X) is proved in [Res07] by purely Geometric
Invariant Theoretic methods; that is, without using Theorem 1.
The first example of face F with cK∨ > 1 is obtained for n = 6.
IJK IJ
etc...
3 Proof of Fulton’s conjecture
Let λ, µ and ν be three partitions (with r parts) such that cν = 1. Let
λµ
us fix n such that n − a is greater or equal to λ , µ and ν . Set I =
1 1 1
{n−a+i−λ : i = 1,···,a} ⊂ {1,···,n}. Similarly, we associate J and
i
K to µ and ν. It is well known that:
cν = cK .
λµ IJ
By Theorem 2, FIJK∨ is a face of codimension one in ∆(n).
Let (A,B,C,A′,B′,C′) ∈ H(r)3 ×H(n−r)3 corresponding to a point
in the relative interior ρIJK∨(FIJK∨). Consider N generic perturbations
(A′,B′,C′) of (A′,B′,C′) ∈ ∆(n−r). Consider now the Hermitian matrix
i i i
A′′ of size r + N(n − r) diagonal by bloc with blocs A, A′,···,A′ ; and
1 N
similarly B′′ and C′′.
Let now, I′′,J′′ and K′′ bethe three subsetsof r+N(n−r)of cardinal r
correspondingto Nλ, Nµ and Nν respectively. It is clear that the image by
ρI′′J′′K′′∨ of the sprectrum of (A′′,B′′,C′′) belongs to ∆(r)×∆(N(n−r)).
By genericity of the matrices A′i, Bi′ and Ci′, this implies that FI′′J′′K′′∨
intersects E(r+N(n−r))++. Now, Corollary 1 allows to conclude.
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References
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N. R.
Universit´e Montpellier II
D´epartement de Math´ematiques
Case courrier 051-Place Eug`ene Bataillon
34095 Montpellier Cedex 5
France
e-mail: [email protected]
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