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**Received date:** September 24, 2009; **Revised date:** April 14, 2010

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We are attempting to give a new proof to the problem of characterization of the support of the product of conjugacy classes in the compact Lie group SU(n) without any reference to the Mehta-Seshadri theorem in algebraic geometry as it was the case in [1].

It is well known that the product of two conjugacy classes in SU(*n*) can be described by a set
of linear inequalities on the Lie algebra of its maximal torus [1], and that these inequalities
are a re-statement of the property of (semi)-stability of certain vector bundles on P(1) with
three (or more) points removed. The proof in [1] depends on a theorem of Mehta-Seshadri
[9] or equivalently can be reformulated in terms of gauge theory of singular
at connections.
For a survey on the case of sum of Hermitian matrices, see the descriptive papers [6,7]. The
purpose of this paper is to give a direct and simple proof of the description of the product
of two conjugacy classes in SU(*n*) which makes no use of the theorem of Mehta-Seshadri
or gauge theory. The main technical tools are an analogue of the Gauss-Bonnet theorem
generally known as the Gauss-Chern formula (see [4]) and a well-known decomposition of
the curvature tensor [8]. These methods are quite elementary and in the course of the proof
we give a clear exposition of some of ideas related to vector bundles on marked Riemann
surfaces.

The main result about the product of two conjugacy classes in SU(*n*) is given by the
following theorem.

**Theorem 1.1.** *Let , β, and
be n-tuples of real numbers in (–1; 1]
such that . Let denote the conjugacy class in SU(n) determined
by the eigenvalues. Then a conjugacy class occurs in the product if and only
if*

*where is an integer and are subsets of the same cardinality . The exact description of the integer and the subsets I, J, and K is given
in Section 4 below.*

In Section 2 we give generalities about vector bundles on marked Riemann surfaces and
their relationship with products of conjugacy classes. Although the proof of the main theorem
makes no use of algebraic geometry of parabolic vector bundles (which was used in [1, 9]), we
give precise denitions which may be useful in giving a direct and simple algebraic proof of
the Mehta-Seshadri theorem which was also proven by Biquard [4] using dierential geometric
methods. In Section 3 we show the (semi)-stability of vector bundles constructed in Section
2 from conjugacy classes by using the Gauss-Chern formula and a decomposition of the
curvature tensor of sub-bundles. Finally in Section 4 we relate the geometric concepts of
Section 3 to our main problem and give an explicit description of the integer and the subsets *I*, *J*, and *K* thereby proving the main theorem. This description involves
quantum multiplication of Schubert cycles.

Let *M** be a compact Riemann surface with *m* marked points or cusps . Throughout the paper the subscript *s* refers to the cusps. Set . We
assume that the Euler characteristic of *M* is negative so that its universal covering space is
the upper half-plane. In addition, we assume that *m* ≥ 1 so that the fundamental group
of *M* is the free group generators. The fundamental group of a
neighborhood of a cusp *p _{s}* is isomorphic to . A neighborhood

Let denote the diagonal matrix with entries and through the paper we
assume that all the eigenvalues are integer numbers in [–1; 1). The given representation
gives a holomorphic vector bundle of rank *n* on *M* as a bre product in the usual manner.
For each *j* let . For dene to be the unique
integer *l* such that. Set

in which *N* is the least integer number such that for (the choice of N is
for the matter of holomorphicity of the map *g*). Note that

(2.1)

Since a holomorphic vector bundle on *M* is holomorphically trivial, it can be extended to
any other vector bundle of rank *n* on . We will describe a specific extension of to and throughout we will only consider this extension. Identifying *U _{j}* with the unit disc with
the origin removed , we glue with by the map

This gives us the extension of to which we call the *standard extension* and denote by if necessary for emphasis. In view of the standard extension we dene the (rst) Chern
class of as .

**Lemma 2.1.** *The line bundle det is holomorphically trivial and consequently the Chern
class of is zero.*

**Proof.** By the above construction the transition functions of are given by

where matrices *E* and are diagonal matrices in U(*n*) with diagonal entries and , and .

Precisely, according to the construction of the trivialization around the point 0 and 1 in the previous step, for each trivialization we consider the diagonalization of the generator of the stabilizer subgroup of the corresponding cusp, so in general we cannot simultaneously diagonalize the two matrix corresponding to the parabolic points 0 and 1. Let

and the matrix is conjugate to the diagonal matrix *β* by a matrix as follows:

So by the construction for the transition function we have

The Chern class of the bundle is equal to the Chern class of the determinant bundle, which is a line bundle with transition function

proving that det is in fact a trivial bundle and hence has vanishing Chern class.

To define the notion of parabolic bundle, at each cusp *p _{j}* we fix a flag subject to the requirement that the subspace is invariant under the action of which
acts as the scalar . Let be a holomorphic sub-bundle with fibre
dimension

The parabolic slope is

Given a holomorphic sub-bundle of rank *r*, one obtains a parabolic structure on *F* as
follows. An ascending flag in the fiber *F _{p}* at marked (cusp) point

those terms for which the inclusion is not strict (note that since the vector space *F _{p}* is
an

A parabolic sub-bundle of *E* is a holomorphic sub-bundle whose parabolic structure
is the one induced from the inclusion. We say the parabolic bundle is parabolic semi-
stable (stable) if for all parabolic sub-bundles . Now we
introduce the necessary tools for dealing with parabolic vector bundle from the differential
geometric viewpoint.

In this section we relate the weights defined in the previous section for parabolic bundle more
intrinsically in a way that these numbers is corresponded to the geometry of the bundle.
Assume that the bundle *E* over the Riemann surface X is parabolic at the cusp point *p* and
equipped with a hermitian metric *h* smooth on *X* –{*p*} and degenerate at *p* and in some
sense which will be made precise later this metric is adaptive with the parabolic structure.

Let be the space of holomorphic structures over *E* or more precisely the space of
operators

Let *A* be the space of *h*-unitary connections which the associated holomorphic structure. In other words, this is the space of *h*-unitary connections which is smooth on *X* –{*p*} whose (0, 1) part is smooth on all of *X*. The corresponding gauge group for the parabolic
bundle *E* is

Similarly the gauge group for hermitian bundle (*E,h*) is defined by

The following definitions explain in what sense a metric is adaptive with the parabolic structure.

**Definition 3.1.** One says that the frame (*e _{i}*), which is a basis for the bundle

**Definition 3.2.** Suppose *r* > 1 and let be a local basis of sections in a neighborhood
of the point *p*. One says that the basis is

(1) adaptive with the parabolic structure of *E* if

for and (*f _{i}*) is a basis for

(2) adaptive with (*E,h*) if it is adaptive with *E* and furthermore it is *h*-orthonormal.

**Definition 3.3.** The hermitian metric *h* on is adaptive if (*E,h*) admits an adaptive
basis according to the previous definition.

**Remark 3.4.** By definition it is clear that the bundle *E* always posses an adaptive basis
and the same statement is valid for hermitian bundle (*E,h*).

Now we bring an example which is illuminating for the above denitions and we will refer to the computation in this example in the next section.

**Example 3.5** (construction of an adaptive metric). Let *z* be a local coordinate for *X* around
*p* and let be a local basis of sections for *E* respecting the flag. One can locally define
a metric for *E* as follows:

In fact this metric can be extended smoothly on *X* –{*p*} and it is clear that the basis is adaptive for (*E,h*).

Furthermore, we assume that *E* is a holomorphic bundle having *E* as its underlying ber bundle and also we assume that the sections (*e _{i}*) are holomorphic sections of E.

The associated Chern connection of the metric *h* can be locally written as

where *α* is the diagonal matrix with coefficients . we use the orthonormal
frame instead of (*e _{i}*) after some ordinary calculation, we obtain the formula

From the above computation it is clear that the curvature of the connection *d ^{h}* vanishes. We say that the holomorphic parabolic bundle is decomposable if it admits a holomorphic
decomposition such that and are holomorphic sub-bundles and they are
equipped with the parabolic structure induced from the parabolic bundle

Now we can describe the notion of parabolicity of a holomorphic bundle in terms of geometry of the bundle itself as follows.

By a parabolic structure for a holomorphic bundle over a point *p* we mean a choice of
an adaptive metric *h* degenerate at *p* and the type of degeneracy at *p* determines the weights
and the flag structure over cusp point *p*. So we can deal with the notion of parabolicity by
using the geometry of the space of singular connections of the bundle *E*.

With this differential geometric viewpoint of parabolicity we bring a theorem which en- lightens the relation between the notion of parabolic degree and the geometry of the holo- morphic bundle which plays the role of Gauss-Bonnet theorem in the context of parabolic bundles (see [4]).

**Theorem 3.6** (Gauss-Chern formula).* Suppose h is an adaptive metric on the holomorphic
bundle and consequently E equipped with a parabolic structure; then for every connection we have*

**Proof.** Since , in terms of the adaptive basis one has the local expression

Let *A _{0}* be a connection on

The differential 1 - form *c* = *A* – *A _{0}* which is defined on

It is easy to see that . Let be a ball of radius around *p* with
boundary by integration over and using Stokes theorem we obtain

Since , we have

when hence by tending to zero we obtain

The right-hand side of the above formula is exactly the definition of the parabolic degree and nally Gauss-Chern formula was proved.

Using this differential geometric viewpoint of parabolic bundle we are ready to pose the main theorem in [4] which characterize the stable (semi-stable) parabolic bundles in a differential geometric fashion. For differential geometric proof of a similar theorem concerning ordinary stability, we refer to [5].

**Theorem 3.7.** *Let be an indecomposable parabolic bundle equipped with an adaptive her-
mitian metric h. The bundle E is parabolic stable if and only if there exists a connection
satisfying*

*Moreover, this connection is unique up to the action of the gauge group.*

*Let and * be two conjugacy classes in the group SU(*n*) and suppose the conjugacy
class occurs in the product of conjugacy classes * and *. Equivalently the identity
matrix occurs in the product we are attempting to characterize all such 's in the
Lie algebra of maximal torus. So in this case we have a representation such that and . Through this
section we choose the integer number *N* so that all the numbers and are
integer numbers.

According to the construction in Section 2 we see that corresponding to this representation
there is a special extended bundle over which we named *. Furthermore, this bundle
is a parabolic bundle with parabolic structure over the cusp points 0, 1, by the definition
posed in Section 2. Now we are attempting to show that the parabolic bundle over is
semi-stable. To this aim first we construct a special connection on as follows.*

As we know from the Section 2 the bundle over is trivial, so it
admits a
flat connection on *M*. Although this connection is not defined on parabolic points 0, 1, and similar to the extension of the bundle on the extended bundle with singularities at
parabolic points 0, 1, and . For this purpose we consider the singular connection :

in the neighborhood around 0 and similarly we define the singular connection in the neighborhood around 1:

and we also consider the singular connection in the neighborhood around the parabolic point .

Now we show that the connections and satisfy the compatibility conditions and therefore we can define a global singular connection on the bundle with singularities at parabolic points 0, 1, and .

To check the compatibility conditions we begin to verify the condition for the two singular connections and ; in fact we should verify that on we have

In Section 2 we see that is the transition function with for the
diagonal matrices *α* and *β*:

For the right-hand side of the above relation, we have

So the compatibility condition for two connections and has been shown for the other two connections , and , ; the compatibility condition can be similarly verified.

Now we check the compatibility condition for the connection with each connection , , and ; for example we show the compatibility condition between the connection and (for the other two connections and , the argument of compatibility is exactly the same).

According to a theorem (see [8]) since the bundle is trivial, it admits a
at structure for which the flat connection on the bundle has the representation . Let
*U _{0}* be an open set containing 0; according to the triviality of the bundle it is evident that
the transition function is equal to .

Now we have to prove that the connection on *U _{0}* and on is compatible on in fact we should verify

by substituting and ; the validity of the above formula is trivial because we have

which is the definition of the singular connection ; thus the compatibility of the connections and was verified. Therefore, all the connections , , , and are compatible
over , so we can dene a global singular connection on the parabolic bundle with
singularities at parabolic points 0, 1, and of the type *α, β*, and *γ,* respectively.

It can easily be seen that the curvature of the connection vanishes and also the connec- tion has zero curvature, so the curvature of the global singular connection introduced above vanishes; hence we have the following theorem.

**Theorem 4.1.*** The parabolic degree of the parabolic bundle is zero.*

**Proof.** According to Theorem 3.6, we have

but we have and consequently

Therefore, by the above argument to a triple (*α*,* β, γ*) of eigenvalues (in which *γ* ccurs in
the product of two fixed conjugacy classes * and *) we associate a flat singular connection
with singularity at 0, 1, and of residues *α, β*, and *γ,* respectively.

The singular flat connection described above induces a singular connection on every
sub-bundle with appropriate residues and which is not necessarily flat. In the following we want to prove that for all such sub-bundles *S* we have; in other words, the bundle is semi-stable.

**Theorem 4.2.** *The parabolic bundle is semi-stable.*

**Proof.** By the argument in this section we see that the bundle admits a singular con-
nection with appropriate singularities at cusp points 0, 1, and moreover by the
construction its curvature is identically zero. Let be a holomorphic sub-bundle of the
parabolic bundle . So we have the exact sequence of holomorphic bundles

where *Q* is the quotient bundle. Using a well-known theorem in dierential geometry for
example we refer to [8]; the unitary connection *A *on *E* described above has the following
shape:

With *A _{F}* and

where is built from *A _{Q}* and

Furthermore, by the construction of this connection we know that the above curvature
matrix, *F*(*A*), is identically zero. Hence, for the curvature of the connection *A _{S}* of the sub-
bundle we have

Now by Gauss-Chern formula proved in Section 3 we have

So we proved and consequently we have . Hence, by the above argument we showed that the bundle is semi-stable.

The direct consequence of this theorem is the following corollary.

**Corollary 4.3.** *The support of the product of two conjugacy classes in SU(n) is contained
in the set of inequalities, of the form where S goes over the sub-bundles of the
parabolic bundle , concerning the semi-stability of the parabolic bundle .*

Notice that the above inequality is equivalent to the inequality, so
for better understanding of the inequalities of this form we should be able to compute the
parabolic degree of the sub-bundle or equivalently we should know the parabolic
weights induced from the bundle to the sub-bundle *S*.

As we mentioned in the definition of semi-stability in the beginning of this paper the
induced weights on the sub-bundle *S* of rank *r* is as follows.

We follow our important case, the parabolic bundle on , in this case we know that *α, β*, and *γ* are the parabolic weights on the cusp points 0, 1, and , respectively. For example we explain the induced weights on the cusp points 0 for the sub-bundle S and for the other
cusp points the argument is exactly the same. To do this let us assume that the flag structure
on the ber over the point 0 in the bundle (recall the denition of parabolicity) is

An ascending flag in the ber *S _{0}* at marked (cusp) point 0 for the sub-bundle

those terms for which the inclusion is not strict (note that since the vector space *S _{0}* is an

So by this explanation the parabolic degree of the sub-bundle *S* is equal to

where

*J _{S}* and

Now we are ready to bring our result about the product of two conjugacy classes in terms of an appropriate set of linear inequalities as a corollary of this section.

**Corollary 4.4.** *The support of the product of two conjugacy classes α and β is the set of
all eigenvalues γ which is necessarily contained in the set of linear inequalities of the form*

*where S goes over the sub-bundles of the parabolic bundle and the subsets I _{S}, J_{S}, K_{S} are
the corresponding indices of the induced parabolic weights for the sub-bundle S as explained
above.*

To see that this set of linear inequalities are also sucient to characterize the product of
two conjugacy classes we can make use of the convexity theorem for Hamiltonian action of
loop groups proved by Meinrenken and Woodward [10], which says that this support, as a
subset of maximal torus, is a convex polytope of maximal dimension. Notice that it can be
easily veried that all boundary hyperplanes dened by the above linear inequalities are in
the support and so by the above-mentioned convexity result we can deduce that the whole
convex set dened by the above (IJK)-inequalities are the exact support of the product of two
conjugacy classes * and *. Note that some of the above inequalities may be redundant;
to see how one can choose an independent set of linear inequalities to describe the support,
we refer to [2,3].

We can also prove the suffciency of the (semi-stability) inequalities by using the corre-
spondence between the moduli of singular flat connections (up to the gauge group action)
and the product of conjugacy classes * and *, which we bring in the following.

**Theorem 4.5.** *Assume that a holomorphic vector bundle E over admits a singular
at
unitary connection (with respect to the degenerate metric) with singularity at 0, 1, and whose residues are α, β, and γ, respectively. Then γ occurs in the product of two conjugacy
classes and *.

**Proof.** According to Theorem 4.2 this bundle is automatically semi-stable. The monodor-
omy of the flat connection gives us a representation of the fundamental group of *M*, namely, . One can easily show that the
condition of the residue around 0 to be *α* is equivalent to the condition (because
locally the connection around 0 has the form ) and similar conditions are satised
for singularities 1 and . So we have a representation *ρ* in which and and consequently *γ*
occurs in the product of two conjugacy classes * and *.

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- W. Fulton. Eigenvalues, invariant factors, highest weights and Schubert calculus.Bull.Amer.Math. Soc. (N.S.),37(2000), 209{249.
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