Multiplicities for tensor products on special linear versus classical groups

There is a natural bijective correspondence between irreducible (algebraic) selfdual representations of the special linear group with those of classical groups. In this paper, using computations done through the LiE software, we compare tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. More precisely, under the natural correspondence of irreducible finite dimensional selfdual representations of SL_2n(ℂ) with those of Spin_2n+1(ℂ) , it is easy to see that if the tensor product of three irreducible representations of Spin_2n+1(ℂ) contains the trivial representation, then so does the tensor product of the corresponding representations of SL_2n(ℂ) . The paper formulates a conjecture in the reverse direction for the pairs (SL_2n(ℂ), Spin_2n+1(ℂ)), (SL_2n+1(ℂ), Sp_2n(ℂ)), and (Spin_2n+2(ℂ), Sp_2n(ℂ)) .