This question concerns character tables. They seem to have so high number of algebraic properties that it seems to be impossible to construct a believable one that is not in fact a character table of any particular group. Question arises:
Are there any character-table-like matricies that are not actual character tables?
More precisely:
Let extended character table mean $n \times n$ matrix $M$ with associated $\mathbb{Z}^n$ vector $v$ (corresponding to class sizes) with following properties:
Let $S$ denote sum of entries of $v$.
- First entry of $v$ is equal to $1$.
- All entries of $v$ are positive
- $v_i$ divides $S$.
- All entries in the matrix $M$ are algebraic integers
- Rows of the matrix obey orthogonality relation as in usual character table (that is $M_{ij}v_j \overline{M_{kj}} = \delta_{ik}$)
- Columns of the matrix obey orthogonality relation as in usual character table (that is $M_{ij}M_{ik} = (v_j)^{-1} S \delta_{jk}$)
- Rows of the matrix are bounded by the first entry (that is $\forall_j |M_{ij}| \leq |M_{i1}|$)
- First column consists of positive integers (edit suggested by @Joppy)
Finally:
Are there any extended character tables that do not arise from a group?
What about this one: $$\begin{pmatrix} 1&1&1&1&1&1\\ 1&1&1&1&-1&-1\\ 1&1&1&-1&1&-1\\ 1&1&1&-1&-1&1\\ 2&2&-2&0&0&0\\ 4&-2&0&0&0&0 \end{pmatrix}?$$ Taken from the paper "Character table sudokus" https://link.springer.com/article/10.1007/s00013-023-01859-w See also the reference therein.