**Background:**

A representation $ \rho: G \to GL(V) $ of a group $ G $ on a (complex) vector space $ V $ is said to be *irreducible* if there are no nontrivial invariant subspaces under the action of $ G $. A tensor product of two irreducible representations $ \rho_i: G \to GL(V_i) $, $i=1,2$, is a representation on the tensor product space $ V_1 \otimes V_2 $, defined by $ \rho_1(g) \otimes \rho_2(g) $ for $ g \in G $.

In some cases, this tensor product representation $ \rho_1 \otimes \rho_2 $ can itself be irreducible. 

**Examples:**

- If $V_1$ or $V_2$ is one-dimensional, then $ \rho_1 \otimes \rho_2 $ is irreducible.

- Let $G$ be the product $ G_1 \times G_2 $. If $ \rho_i $ is an irreducible representation of $ G_i $, then the representation  $\rho_1 \otimes \rho_2 = (\rho_1 \otimes 1) \otimes (1 \otimes \rho_2)$ is irreducible for $ G_1 \times G_2 $. 

To avoid such straightforward examples, let us assume that $G$ is a non-abelian finite simple group and $\rho_i \neq 1 $. Surprisingly, such phenomenon is still possible in this case, but quite rare. It occurs just on seven such groups of order less than $2\times 10^7$, namely, $PSp(4,3)$, $M_{12}$, $A_9$, $PSp(6,2)$, $M_{23}$, $PSU(5,2)$, $2F(4,2)'$. See the computation in Appendix. 

More specifically, for the alternating groups $A_n$,  with $5 \le n \le 20$, this phenomenon occurs just for $n=9,16$, with $(\dim(V_1),\dim(V_2))=(8,21), (15,12012)$, respectively. 

**Question**: For which non-abelian finite simple groups this phenomenon occurs? more specifically, for which alternating groups $A_n$? if and only if $n$ is a square? and $\dim(V_1) = n-1$? 

My laptop was not able to check up to $n=25$. 

_________
**Computations**  

We know that we can avoid $PSL(2,q)$, see [here][1]. 

    it:=SimpleGroupsIterator(60,20000000);; for g in it do N:=Name(g);; if Size(N)<6 or List([1..6],i->N[i]) <> "PSL(2," then Print(Name(g),"\n",Order(g),"\n");; Phenomenon(g);; fi; od;

    PSp(4,3), M12, A9, PSp(6,2), M23, PSU(5,2), 2F(4,2)'

Computation for the alternating group $A_n$,  for $5 \le n \le 20$:


    for n in [5..20] do Print(n,"\n");; g:=AlternatingGroup(n);; Phenomenon(g);; od;

    A9, A16

**Script**

    Phenomenon:=function(g)
    	local irr,r,L,i,j,k;
    	irr:=Irr(g);
    	r:=Size(irr);
    	for i in [2..r] do
    		for j in [i..r] do
    			L:=List([1..r],k->ScalarProduct(irr[i]*irr[j],irr[k]));;
    			if Number(L,i-> i=0)=r-1 and Number(L,i-> i=1)=1 then 
    				Print([i,j],"\n");; 
    			fi;
    		od;
    	od;
    end;;


  [1]: https://doi.org/10.1142/S0129167X23500301