/Actual matrices/ : Given a particular representation R of a group, you want to break it up into a direct sum of irreducible representations, R = R0 (+) R1 (+)

/Actual matrices/ : Given a particular representation R of a group, you want to break it up into a direct sum of irreducible representations, R = R0 (+) R1 (+) ... (+) Rk. Based on the character theory, you can write down this decomposition without actual matrices... But suppose you want some actual matrixes to play with. The 'direct sum' statement is the same as saying that there's a basis for R in which the group action is block diagonal, with the block elements being something from R0, R1, ..., Rk respectively. The 'concrete' requirement probably means you want to find a particular change-of-basis operation for your preferred basis of R and the block diagonal one; ie, find P such that: P R(g) P^t = R0(g) (+) ... (+) Rk(g) for every element g in the group. My lazy answer, then, is that if you have a concrete (eg, in-matrices) knowledge of R and each of the irreps, you can just solve the (many) linear equations to find P. (Remember that each g gives us another matrix equation here, which all hold simultaneously for P, so this is massively overdetermined.) Another way would be to construct projections pi from R into each of the irreducible representations, and rearrange R as pi_inverse(Image) (+) Kernel(pi) for each of these projections.