# Monster coefficients

For three irreducible characters $\phi,\psi,\rho$ of a finite group $G$ define the Kronecker multiplicities as: $$g(\phi,\psi,\rho) = \langle \phi,\psi\cdot\rho\rangle$$ where $$\langle \chi,\eta\rangle = \frac{1}{|G|}\sum_{x\in G} \chi(x)\, \overline{\eta(x)}$$ and $[\psi\cdot\rho] (x) = \psi(x) \rho(x)$ is the usual product.

I am interested in Kronecker multiplicities for the Monster group $M$. While the group is large, there are only 194 conjugacy classes.

$$(1) \qquad \max_{\phi,\psi,\rho} g(\phi,\psi,\rho)$$

$$(2) \qquad \sum_{\phi,\psi,\rho} g(\phi,\psi,\rho)$$ $$(3) \qquad \sum_{\phi,\psi,\rho} g(\phi,\psi,\rho)^2$$

These sums are over all triples of irreducible characters, but because of the symmetries only about 1/6 of them need to be computed to get the answer. If you can do this, I would also be curious about the specific characters maximizing (1).

The computation is beyond my computer algebra skills, but I know that GAP has the whole character table of $M$ ready to use.

• One can try to ping @user:2820 (Derek Holt), he might know. Feb 27 '18 at 0:55

In GAP, you could simply iterate in a triple loop over $\phi,\psi,\rho$, calculate the $g$-values and find maximum and sum values:

ct:=CharacterTable("M");
irrs:=Irr(ct);

m:=0; s:=0;q:=0; # max, sum, sumsquare

for rho in irrs do
for psi in irrs do
ten:=rho*psi; # tensor product
for phi in irrs do
g:=ScalarProduct(phi,ten);
if g>m then m:=g;fi;
s:=s+g;
q:=q+g^2;
od;
od;
od;


Afterwards look at the values of m, s, and q. Unless I have mistyped something, the results I get are (for the monster group):

1. Maximum is $21458051228477513179513856=2^{10}\cdot281\cdot443\cdot599\cdot6571\cdot42768299767$

2. Sum is $247017097351847432984363535932$ (Thank you, @James for the correction)

3. Sum-Squares is $808017424794512875894769468067441075690144312450960558$ (ditto corrected, also typo fixed)

• I computed the same maximum, but got different values for the sum and sum of squares. My value for the sum is $247017097351847432984363535932$ and, for the sum of squares is $808017424794512875894769468067441075690144312450960558$. If I'm not mistaken, I think your code should have added your "g" and "g^2" rather than "m" and "m^2\$ to "s" and "q" respectively. Feb 28 '18 at 16:26
• @James, Oops, you're completely right. I'll fix it. Feb 28 '18 at 17:18
• I was just about to report that I modified your GAP code and ran it and obtained the same results that I had seen from my own code. It is perhaps useful to know that the results can be replicated independently in two different CASs. Feb 28 '18 at 17:32