Why the equation in Burnside's lemma has only one solution

Question

Burnside's lemma (for finite group) tells us that the square sum of dimensions over all inequivalent irreducible representations is equal to the order of the group, i.e.

$$ d_1^2 + d_2^2 + \cdots + d_r^2 = |G| $$

where $d_i$ is the dimension of representation $D^{(i)}(g)$. It seems that for all finite groups, this equation has only one solution (for $d_i$). Is this true and how can I prove it or find a counter-example?

Answer 1

If a finite group $G$ has a representation with $d_i=2$ and $d_j=9$ then one can replace these by $6$ and $7$ without affecting the sum of the $d_i^2$s. So the equation $d_1^2 + d_2^2 + \cdots + d_r^2 = |G|$ may have multiple solutions in positive integers.

Answer 2

$$8 \times 1^2 + 2 \times 2^2 + 4^2 = 4\times 1^2 + 7 \times 2^2 = 32.$$ There are groups of order $32$ (numbers $6$ and $18$ in the database) with these character degrees.