I would like to better understand the relationship between character tables, representations, and "symmetry arguments" when discussing "symmetric" solutions to systems like:
Consider a set of vectors we can label with the elements of some group G, that have a "symmetric" weighted sum of zero: $$\sum_{\sigma \in G} \text{weight}(\sigma) \ v_\sigma = 0$$
I am interested in the general approach, but for sake of discussion:
Consider a set of 6 vectors we can label with the elements of $S_3$. I'll write these elements as [123],[132],[312],[321],[231],[213].
The only kind of symmetry argument I easily understand currently is a parity argument. That is, if vectors obey: v_[123] = - v_[132], and similarly for other parity swaps, then I can clearly argue: $$\sum_{\sigma \in S_3} \text{parity}(\sigma) \ v_\sigma = 0$$
But that involved simple pairwise cancellation: v_[123] + v_[132] = 0. In otherwords it was due to a "simpler" symmetry in the set: pairing them up and labeling the pairs by elements of Z_2.
There are probably other "symmetric" solutions that cannot be broken down this way, where some "term" in v_[123] that is cancelled by -v_[132], such that the full "weighted sum" above is still zero but pairwise v_[123], v_[132] do not sum to zero.
How can I systematically explore these possibilities?
And I definitely am at a loss when I see there are representations of dimension > 1. For example S_3 has dimension 2 reps, what is that saying for how the vectors should be related or how I weight them in a sum? Maybe complex numbers? But then what about groups with dimension 3 reps?
I'm sure this is a large and deep topic, but I am primarily interested in how I can look at a character table and figure out what different "symmetric" possible solutions to such a sum may look like.
I'm currently trying to solve a problem with polynomials with a bizarre kind of symmetry, and I think they sum to zero given the right kind of "symmetric sum". But beyond a parity argument, I have no idea what these could look like, so I have no idea what I should be trying to prove about the relationship of these polynomials. An engineering friend suggested I learn about symmetry tables and while this definitely looks like a useful path, I'm struggling to figure out how to use this information.