Multiplicities for tensor products on special linear versus classical groups

Abstract

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 $\mathrm{SL}_{2n}({\mathbb {C}})$ with those of $\mathrm{Spin}_{2n+1}({\mathbb {C}})$, it is easy to see that if the tensor product of three irreducible representations of $\mathrm{Spin}_{2n+1}({\mathbb {C}})$ contains the trivial representation, then so does the tensor product of the corresponding representations of $\mathrm{SL}_{2n}({\mathbb {C}})$. The paper formulates a conjecture in the reverse direction for the pairs $(\mathrm{SL}_{2n}({\mathbb {C}}), \mathrm{Spin}_{2n+1}({\mathbb {C}})), (\mathrm{SL}_{2n+1}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}})),$ and $(\mathrm{Spin}_{2n+2}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}}))$.