This question is about group representation theory but might be a bit specific.
My question is from the following book:
Group Theoretical Methods and Their Applications (by E. Stiefel, A. Fässler) p.110 (If you know group representation theory, you can directly read Ch.5)
Please see (5.29): $$D(s)b^i_{\rho} = \sum_{\sigma=1}^{n_j} d^{(j)}_{\sigma \rho} b^i_{\sigma}$$
It looks like $d^{(j)}_{\sigma \rho} $ is a vector and $ b^i_{\sigma}$ is also a vector, basis vectors of $V^i_j$. Then how to do this multiplication?
I am sorry I cannot provide the whole background about this question since there are too many stuffs here. So if you have read this book before, please help me.