Lmov Conjecture And Representation Theory

There are two main parts in this paper. First, we find and solve a repre-sentation problem. It arise from algebraic topology and differential topology. It comes from a discussion with my teachers with the class time. At the beginning, we have a great metric on the fiber of our fiber bundle(always is fundamental or fundamental symmetric). then form algebraic topology, we can deduce a new metric on the fiber bundle as the tensor of the metrics of fiber(fundamental) and base space(arbitrary). If there exists a metric (subrepresentation) on the fiber bundle? This is the crucial point of the problem. We want a certain answer but failed. Finally, by using lie algebra's representation theory, we find some equal conditions, which is not hard to reach in geometric environment. We can put it here.First, we set the problem simply under lie algebra's condition: Problem:under the condition with lie algebra sln(C)下: assume V is a fundamental representation of sln(C), W is another arbitrary rep-resentation of sln(C)的. our question is: When does there exist a subrepresentation of slnC, we denote it as P such that:symnV (?) W≈symnV (?) PWe use Littleword relation in the process of solving this problem, and turn the problem into solving a not so complicated equations. Littleword(Joe.Harris[JH] P225):Consider the tensor representation ofΓa1,…,an-1 and symkV =Γk,0,…,0。then we can denote its irreducible sum decomposition as:Γa1,…,an-1(?)Γk,0,…,0=(?)Γb1the subrepresentation of this tensorΓb1,…,bn-1 should satisfy following equations: exist n nonegative integers c1,…,Cn such that: ABSTRACT 4 |b1=a1+c1-c2 |b2=a2+c2-c3 |bi=ai+ci-ci+1 (1) |bn-1=an-1+cn-1-cn∑i=1n ci=kAt the same time.we solve the two dimension condition with another method, and find the general method match with it very well. Then,as the second main part of this paper,we give a review about the LOMV conjucture. At that part,I starts with basic knowledge of quantum group in-variants and end with the introduction about LOMV conjucture.The procedure would reflect how I approach this conjucture,and of course,I has some extended condition about some theory which is related to lie alglebra sp and so.Finally,by using isomorphic relation of lie algebras,we get some theory about polynomial invariants under the low dimension.