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I realize that this question is very open-ended since it's not entirely clear what a "valid" character table is.

I would like to know whether creating a character table that has all of the required properties (such as row/column orthogonality, contains a trivial character, etc.) implies that there exists a group with corresponding modules and conjugacy classes.

(Edit) Additional context: I think my question could be simplified into as follows:

Does there exist a set of properties of a character table that allows us to find all finite groups by constructing character tables with those properties? Is this set different to the set of properties that a character table must meet when constructing it from some group?

Dawid
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Feb 25 '23 at 13:51
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    Is this a valid character table?

    \begin{array}{c|cc} &C_1&C_2\\hline \chi_1&1&1\\chi_2&2&-\frac12\end{array}

    There's no group with that table (as it would have to have $1^2+2^2=5$ elements and the only group with $5$ elements is abelian).

     – joriki Feb 25 '23 at 14:25
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    Your edit doesn't look like a simplification to me. I think I understood the original question (up to the uncertainty of what exactly is a "valid character table") – I don't understand the edit. How does a set of properties of a character table allow you to find all finite groups? – joriki Feb 25 '23 at 14:32
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    @joriki I believe fractions cannot appear in a character table hence this would not be a valid-counter example. I will try to explain this further. We know that for any finite group we can find a corresponding character table. My question is whether for any "valid" character table we can find a corresponding group. The main issue is how do I define "valid". We know that there are certain properties of a character table that are always required (row/column orthogonality etc), but are those properties only necessary or also sufficient? Does there exist a set of properties that is sufficient? – Dawid Feb 25 '23 at 14:41
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    I guess a good start would be to begin with square tables satsifying row/column orthogonality such that all the values are algebraic integers, with the first row filled with $1$s and the first column featuring integers such that each of them divides the sum of the squares? (I've just started character theory so maybe I am missing other """obvious""" characteristics of character tables?) – Bruno B Feb 25 '23 at 14:47
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    The first major obstacle being that it's apparently hard/impossible currently to even know if which sums of squares correspond to a group to begin with, see the comments here: https://math.stackexchange.com/q/4303663/1104384 (depending on what you're willing to consider valid this may already answer your question) – Bruno B Feb 25 '23 at 14:52
  • @BrunoB Thanks, this is exactly what I was looking for! – Dawid Feb 25 '23 at 14:59
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    see https://math.stackexchange.com/questions/4241366/validating-a-character-table-for-a-given-finite-group/4243041 – Brauer Suzuki Feb 25 '23 at 18:56

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In his famous report for the ICM, Richard Brauer proposed a number of questions on representations of finite groups. His Problem 6 asks to give necessary and sufficient criteria on a complex matrix to be a character table of a group. I think there is no satisfactory answer as of today. To give you a non-trivial challenge: Is the following matrix a character table? $$\begin{pmatrix} 1&1&1&1&1&1\\ 1&1&1&1&-1&-1\\ 1&1&1&-1&1&-1\\ 1&1&1&-1&-1&1\\ 2&2&-2&0&0&0\\ 4&-2&0&0&0&0 \end{pmatrix}$$

(the answer is in my recent paper on “character table sudokus”).

Brauer Suzuki
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