Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$

Let the **tensor multiplicity** of $G$ be $m(G):= \max_{i,j,k} (n_{i,j}^k)$; for ex. $m(A_5) = 2$ and $m(A_6) = 3$.

*Theorem*: Let $q > 5$ be a prime-power, then $$m(\mathrm{PSL}(2,q)) = \left\{
\begin{array}{ll}
2 & \text{ if } q \text{ even,} \\
3 & \text{ if } q \text{ odd.}
\end{array}
\right.$$
*proof*: see (for example) the generic computation of $(n_{i,j}^k)$ for $\mathrm{PSL}(2,q)$ in my talk at 30:00. $\square$

This post is about the converse of above theorem.

Let $G$ be a non-abelian finite simple group with $m(G) \le 3$.

**Question**: Is it true that $G \simeq \mathrm{PSL}(2,q)$ for some prime-power $q$?

It was checked by GAP for $|G|<10^7$ (see Appendix).

*Remark*: The classification below suggests that the family $(\mathrm{PSL}(2,q))$ is the only one, among the usual infinite families of (non-abelian) finite simple groups, where the tensor multiplicity is bounded above. This would provide an other interesting characterization of this family.

**Appendix**

We improved our way to check as suggested by Mikko Korhonen in comment. Let $G$ be a finite group, $d_+(G)$ be the maximum among the degrees of complex irreducible characters of $G$, and $\Sigma(G)$ their sum. They are exactly $27$ non-abelian finite simple groups $|G|<10^7$ not isomorphic to some $\mathrm{PSL(2,q)}$. The classification below lists them ordered according to $d_+^2(G)/\Sigma(G)$, which is less than or equal to $m(G)$.

*Computation*

Each element of the list reads as $d_+^2(G)/\Sigma(G)$, its numerical approximation, $G$ and $|G|$.

```
gap> classification(10000000);
[ [ 256/63, 4.06349, PSU(3,3), 6048 ],
[ 1125/208, 5.40865, PSU(3,4), 62400 ],
[ 13/2, 6.5, PSL(3,3), 5616 ],
[ 18432/2107, 8.74798, PSU(3,7), 5663616 ],
[ 175/18, 9.72222, A7, 2520 ],
[ 2048/203, 10.0887, PSL(3,4), 20160 ],
[ 175/16, 10.9375, A8, 20160 ],
[ 279/25, 11.16, PSL(3,5), 372000 ],
[ 729/64, 11.3906, PSp(4,3), 25920 ],
[ 3025/218, 13.8761, M11, 7920 ],
[ 8281/484, 17.1095, Sz(8), 29120 ],
[ 10368/553, 18.7486, PSU(3,5), 126000 ],
[ 7225/271, 26.6605, PSp(4,4), 979200 ],
[ 321489/11210, 28.6788, PSU(3,8), 5515776 ],
[ 43681/1464, 29.8367, J_1, 175560 ],
[ 704/21, 33.5238, M12, 95040 ],
[ 23328/683, 34.1552, A9, 181440 ],
[ 5472/155, 35.3032, PSL(3,7), 1876896 ],
[ 28224/709, 39.8082, J_2, 604800 ],
[ 16384/319, 51.3605, PSp(6,2), 1451520 ],
[ 321489/5356, 60.0241, A10, 1814400 ],
[ 152100/2519, 60.3811, PSp(4,5), 4680000 ],
[ 21175/258, 82.0736, M22, 443520 ],
[ 173056/1875, 92.2965, G(2, 3), 4245696 ],
[ 20800/207, 100.483, PSL(4,3), 6065280 ],
[ 775/7, 110.714, PSL(5,2), 9999360 ],
[ 28672/227, 126.308, PSU(4,3), 3265920 ] ]
```

Below is the alternative classification suggested by Goeff Robinson in comment, considering $d_+(G)/c(G)$, with $c(G)$ the class number of $G$:

```
gap> classification2(10000000);
[ [ 16/7, 2.28571, PSU(3,3), 6048 ],
[ 13/4, 3.25, PSL(3,3), 5616 ],
[ 75/22, 3.40909, PSU(3,4), 62400 ],
[ 35/9, 3.88889, A7, 2520 ],
[ 81/20, 4.05, PSp(4,3), 25920 ],
[ 5, 5., A8, 20160 ],
[ 11/2, 5.5, M11, 7920 ],
[ 31/5, 6.2, PSL(3,5), 372000 ],
[ 32/5, 6.4, PSL(3,4), 20160 ],
[ 192/29, 6.62069, PSU(3,7), 5663616 ],
[ 91/11, 8.27273, Sz(8), 29120 ],
[ 72/7, 10.2857, PSU(3,5), 126000 ],
[ 176/15, 11.7333, M12, 95040 ],
[ 12, 12., A9, 181440 ],
[ 340/27, 12.5926, PSp(4,4), 979200 ],
[ 209/15, 13.9333, J_1, 175560 ],
[ 16, 16., J_2, 604800 ],
[ 256/15, 17.0667, PSp(6,2), 1451520 ],
[ 81/4, 20.25, PSU(3,8), 5515776 ],
[ 228/11, 20.7273, PSL(3,7), 1876896 ],
[ 390/17, 22.9412, PSp(4,5), 4680000 ],
[ 189/8, 23.625, A10, 1814400 ],
[ 385/12, 32.0833, M22, 443520 ],
[ 1040/29, 35.8621, PSL(4,3), 6065280 ],
[ 832/23, 36.1739, G(2, 3), 4245696 ],
[ 224/5, 44.8, PSU(4,3), 3265920 ],
[ 1240/27, 45.9259, PSL(5,2), 9999360 ] ]
```

*Code*

```
classification:=function(n) #for Geoff way: classification2:=function(n)
local it,LL,g,A,L,l,dmax,S,c,cc;
it:=SimpleGroupsIterator(2520,n);; #2520=|A_7|
LL:=[];;
for g in it do
A:=StructureDescription(g);;
if Length(A)<6 or List([1..6],i->A[i])<>"PSL(2," then #PSL(2,q) excluded
L:=CharacterDegrees(g);;
l:=Length(L);;
dmax:=L[l][1];;
S:=Sum(L,i->i[1]*i[2]); #Goeff way: Sum(L,i->i[2]);
cc:=dmax^2/S;;
Add(LL,[cc,Float(cc),g,Order(g)]);;
fi;
od;
Sort(LL);
return LL;
end;;
```

Recall that $A_7$ is the smallest non-abelian finite simple group not isomorphic to some $\mathrm{PSL}(2,q)$.