Unipotency Of Free Group Generated By Two Elements Of Primitive Elements Unipotented

Representation theory of group is one of developing prompt and active branchof mathematics in resent years, and is one of main current direction in algebraicresearch. Representation theory of group has much practical application in quantummechanics and crystal structure and other fields, because it makes complexcalculation simplify, analyses much character to determine their nature, so far as topredetermine development tendency of physical and chemical process.In what respect of study of Lie group and theory of Character, representationtheory of group developed sufficiently. Theory of algebraic group including study ofLie group doesn’t proclaim properties of linear representation of group completely,so theory of combined group becomes a very important tool in researching lineargroup. Study of unipotency belongs to interdisciplinary research of theory ofcombined group and representation.In this paper, we research the necessary and sufficient conditions for unipotencyof free group generated by two elements, when each image of primitive elements inthe linear representation is unipotent matrix.Up to now, studying of this problem refers to two methods: Firstly, byincreasing dimensions of representation of group from low to high, find out generalproof for all dimensions no more than certain fixed integer. Secondly, when thedimensions of represent of group isn’t fixed, as Jordan canonical form we consider acertain kind of matrix canonical form, and get general rule through discuss differentforms. At present, we get the best conclusion that is achieved7dimensions ofrepresentation by first method.This paper mainly research two particular cases when the dimensions ofrepresent of group is8:1) Normalized form of some generator has only one Jordan canonical form.2) Normalized form of some generator has thatdiag (J7,1). Proving the necessary and sufficient conditions for unipotency of free groupgenerated by two elements is that all images of primitive elements are unipotent,when satisfies one of the above conditions.