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].
Highlights
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-stability of certain vector bundles on CP (1) with three points removed
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 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
Summary
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 CP (1) with three (or more) points removed. 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.
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