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On the product of conjugacy classes in unitary group and singular connections | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On the product of conjugacy classes in unitary group and singular connections

Jafar SHAFFAF

Department of Mathematical Sciences, Shahid Beheshti University, G.C., P.O. Box 1983963113, Tehran, Iran
E-mail: j sha [email protected]

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

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Abstract

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].

1 Introduction

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

image

where image is an integer and image are subsets of the same cardinality image. The exact description of the integer image 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 de nitions 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 di erential 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 image and the subsets I, J, and K thereby proving the main theorem. This description involves quantum multiplication of Schubert cycles.

2 Generalities on vector bundles on marked Riemann surfaces

Let M* be a compact Riemann surface with m marked points or cusps image. Throughout the paper the subscript s refers to the cusps. Set image. We assume that the Euler characteristic of M is negative so that its universal covering space is the upper half-planeimage. In addition, we assume that m ≥ 1 so that the fundamental group of M is the free group image generators. The fundamental group of a neighborhood of a cusp ps is isomorphic to image. A neighborhood Us of ps is uniformized by the subset image for some large image and the action of the local fundamental group is by translation image .Let image and note that for imageimage is conjugate in U(n) to a matrix of the form

image

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

image

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

image (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 image. We will describe a specific extension of image to image and throughout we will only consider this extension. Identifying Uj with the unit disc with the origin removed image, we glue image with image by the map

image

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

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

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

image

where matrices E and image are diagonal matrices in U(n) with diagonal entriesimage and image, and image.

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

image

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

image

So by the construction for the transition function image we have

image

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

image

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

To define the notion of parabolic bundle, at each cusp pj we fix a flag image subject to the requirement that the subspace image is invariant under the action of image which acts as the scalar image. Let image be a holomorphic sub-bundle with fibre dimension r. Define the integersimage as follows: a1 is the smallest integer such that image; and image is the smallest integer such that imageimage. Define the parabolic degree of image as

image

The parabolic slope image is

image

Given a holomorphic sub-bundle image of rank r, one obtains a parabolic structure on F as follows. An ascending flag in the fiber Fp at marked (cusp) point p is obtained by removing from

image

those terms for which the inclusion is not strict (note that since the vector space Fp is an r-dimensional vector space, exactly r inclusions of the above sequence of inclusions are strict). The parabolic weights for F are image, where kj is the minimal index such thatimage where image.

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

3 Singular Gauss-Chern formula

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

image

Let A be the space of h-unitary connections which the associated holomorphic structureimage. 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

image

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

image

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

Definition 3.1. One says that the frame (ei), which is a basis for the bundle E at p, respects the flag structure over p if it is a image local basis in a neighborhood of p for the bundle E and furthermore image is generated by image.

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

(1) adaptive with the parabolic structure of E if

image

for image and (fi) is a basis for E at p respecting the flag;

(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 image 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 de nitions 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 image be a local basis of sections for E respecting the flag. One can locally define a metric for E as follows:

image

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

Furthermore, we assume that E is a holomorphic bundle having E as its underlying image ber bundle and also we assume that the sections (ei) are holomorphic sections of E.

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

image

where α is the diagonal matrix with coefficients image. we use the orthonormal frame image instead of (ei) after some ordinary calculation, we obtain the formula

image

From the above computation it is clear that the curvature of the connection dh vanishes. We say that the holomorphic parabolic bundle image is decomposable if it admits a holomorphic decomposition image such that image and image are holomorphic sub-bundles and they are equipped with the parabolic structure induced from the parabolic bundle E and furthermore the union of weights of the induced parabolic structures on image and image is equal to the weights of the parabolic bundle image.

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 image 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 image 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 image and consequently E equipped with a parabolic structure; then for every connection image we have

image

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

image

Let A0 be a connection on E smooth on all of X; then by a similar computation as in the example above one obtains

image

The differential 1 - form c = AA0 which is defined on X –{p} and has values in End(E) has the local expression

image

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

image

Since image, we have

image

when imagehence by tending image to zero we obtain

image

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 image 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 image satisfying

image

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

4 Support of the product of two conjugacy classes

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

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

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

image

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

image

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

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

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

image

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

image

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

image

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

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

According to a theorem (see [8]) since the bundle image is trivial, it admits a at structure image for which the flat connection image on the bundle image has the representation image. Let U0 be an open set containing 0; according to the triviality of the bundle image it is evident that the transition function image is equal to image.

Now we have to prove that the connection image on U0 and image on image is compatible on image in fact we should verify

image

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

image

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

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

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

Proof. According to Theorem 3.6, we have

image

but we have image and consequently image

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 image and image) we associate a flat singular connection with singularity at 0, 1, and imageof residues α, β, and γ, respectively.

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

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

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

image

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

image

With AF and AQ connections on F and Q and β in image. For the corresponding curvature matrix we have

image

where image is built from AQ and AS and the quadratic terms have a definite sign. For convenience normalize so thatimage.

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 AS of the sub- bundle image we have

image

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

image

So we proved image and consequently we have image. Hence, by the above argument we showed that the bundle image 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 image where S goes over the sub-bundles of the parabolic bundle image, concerning the semi-stability of the parabolic bundle image.

Notice that the above inequality image is equivalent to the inequalityimage, so for better understanding of the inequalities of this form we should be able to compute the parabolic degree of the sub-bundle image or equivalently we should know the parabolic weights induced from the bundle image 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 image on image, in this case we know that α, β, and γ are the parabolic weights on the cusp points 0, 1, and image, 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 image (recall the de nition of parabolicity) is

image

An ascending flag in the ber S0 at marked (cusp) point 0 for the sub-bundle S is obtained by removing from

image

those terms for which the inclusion is not strict (note that since the vector space S0 is an r- dimensional vector space, exactly r inclusions of the above sequence of inclusions are strict). The parabolic weights for S over 0 are numbersimage, where kj is the minimal index such that image.

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

image

where

image

JS and KS are de ned similarly.

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

image

where S goes over the sub-bundles of the parabolic bundle imageand the subsets IS, JS, KS 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 veri ed that all boundary hyperplanes de ned 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 de ned by the above (IJK)-inequalities are the exact support of the product of two conjugacy classes image and image. 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 image and image, which we bring in the following.

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

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

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