Is there any explicit decomposition of tensor product of two finite dimensional irreducible modules of simple lie algebras whose highest weights are same?



tensor product of two irreducibles having same maximal weight
The two modules in the question are actually isomorphic, since highest weights determine irreducibles. Anyway, the answer to your question is that there is usually no explicit decomposition known. Of course, there are various algorithmic approaches to tensor product decomposition, but the process tends to get very long. From the viewpoint of the old Brauer method, it's easiest to handle a tensor product when one highest weight is much bigger than the other, whereas the case you consider is rather hard.


For some perspective on the relatively small highest weight $\rho$ (half-sum of positive roots, or sum of fundamental dominant weights), it's worth looking into one of Kostant's papers: *Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ρ*, 
Selecta Math. (N.S.) 2 (1996), no. 1, 43–91.


[In a more positive direction, if the given module happens to be self-dual, then the trivial module occurs precisely once as a summand of the module tensored with itself.]


