TITLE: Characterization of the family of simple groups PSL(2,q) by tensor multiplicity
QUESTION [9 upvotes]: 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 $\mathrm{PSL}(2,q)$.

REPLY [5 votes]: The check of $\textrm{max}\{\textrm{deg}(\chi)\mid \chi \in \textrm{Irr}(G)\} > 3   k$ could be extended to more finite simple groups (including all sporadics) by using the character tables which are available in GAP:
simpnames:= AllCharacterTableNames( IsSimple, true, IsAbelian, false);
LL := [];;                      
for nam in simpnames do  
  Print(nam," ");        
  t := CharacterTable(nam);   
  d := Maximum(List(Irr(t),Degree));  
  k := NrConjugacyClasses(t);    
  Print([d,k,Float(d/k)],"\n");  
  Add(LL, [d/k,Float(d/k),nam, Size(t),d,k]);
od;                         
Sort(LL, {a,b} -> a[4] <= b[4]);   
Filtered(LL, a-> a[1] <= 3);

From the result I would guess the following statement:
G is non-abelian simple with max. character degree $\leq 3k$ iff  G = PSU(3,3) or PSL(2,q).
Sketch of proof:

G sporadic: checked above (GAP, resp. ATLAS)

G alternating: (note that $A_5\cong PSL(2,4)$ and $A_6 \cong PSL(2,9)$ are among the exceptions) Symmetric group $S_n$ has representation for partition
(2,2,2, ...., 2,2 or 1)
and the hook formula and Stirling approximation of $n!$ yield that
its degree grows with $n$ like $e^n$, while the number of conjugacy classes
of $S_n$ (= number of partitions of $n$) grows like $e^{const \sqrt{n}}$.
(The numbers for the alternating group differ at most by a factor $2$).

G of Lie type of rank l:

they always have the Steinberg representation of degree $q^{N}$ where $N$ is the number of postive roots of $G$,  which is roughly $N \sim l^2$
their number of conjugacy classes is bounded by $< c q^l$ for some constant c

(so only rank $l=1$, $PSL_2(q)$, or very small $l$ and $q$ will give exceptions)


So, asymptotically the statement looks ok, one needs to be a bit more precise to get the list of small exceptions.