/sci/ - Science & Math

>>12475497
>Can you recommend a book on representation theory
Not really, sorry. I approach it from the group algebra direction pretty much always, so I can't really recommend anything for sure. A book I sometimes use for reference is Burrow - Representation Theory of Finite Groups, and it is old but good.
>Like 3x3 matrix are a module od dimension 3, 7x7 matrix representation (of the same group) are a module of dimension 7?
I mean the number of rows (and/or columns, square matrices). Let's say we have the field of just 2 elements, 0 and 1. How many 2x2 matrices can we have? 16 is the total, and then we need to have non-zero determinants, which will reduce that to something like 4(ish) possibilities. If we would now try to send the elements from the cyclic group of 16 elements to those matrices, that would necessarily force distinct elements to be sent to the same matrix, and so on.
>Also did you say there is a specific module of some dimension with kernel=0 and lower dimension is a subspace and higher dimension doesn't have new information?
Let's just think of them as vector spaces. If we can get an injection into the automorphisms of an n-dimensional space from our group, then there is nothing new to be learned by mapping our group to the automorphisms of an (n+m)-dimensional space for any m>0, as the image of will then be those matrices in which the nxn top left part is the same, the rest of the main diagonal are 1's and all the other entries 0. That's what I meant. If we get rid of the redundant part, then (I think) this will give the minimal faithful representation of your group, but I'm not too familiar with this matrix formalism to say for sure.
>representation was just "some way" to write down a group or algebra element in a concrete way.
Yes. Any group can be turned more concrete by embedding it into a symmetric group. Similarly, these representations can be used to turn an arbitrary group into matrices.

Really hitting the character limit lately.