# Group Representations and Young Tableaux

1. May 17, 2006

### AKG

What are good resources on Young diagrams and tableaux for representations of the permutation groups Sn and the unitary groups U(n) of n x n unitary matrices?

2. May 17, 2006

### matt grime

William Fulton has an entire book on S_n (and matrix group) stuff. It's an LMS student text called something like 'Young Tableaux'

3. May 17, 2006

### AKG

Thanks, I'll look into that. My professor says that people use Young tableaux to figure out the irreducible representations of the symmetric groups and the unitary groups. However, he says that although people know the applications of Young tableaux, and have proven that they work, there's not much of an understanding as to why they work. As a project, he has asked me to try and figure it out, if possible. He recommended a book by an author named Sagan (forget the title, or the author's first name, but I have the book on hold). Do any resources come to mind that would help with this particular thing, i.e. figuring out why Young tableaux work? Does the Fulton book do this?

4. May 17, 2006

### George Jones

Staff Emeritus
I went looking for the text for a couple of courses that I took as a student and found it. This book was old when I used it, and is even older now. It looks much denser than I remember and may not be of much use to you. In any case, it is now in the http://www.ima.umn.edu/~miller/symmetrygroups.html" [Broken]. Chapters 4 and 9 may relevant.

Regards,
George

Last edited by a moderator: May 2, 2017
5. May 18, 2006

### matt grime

Hmm, I hate these 'why' questions. I'd say we understand rather well why Y.T. parametrize the reps of S_n, in fact we know a hell of a lot about them (they also parametrize modular reps and one can do many things involving hook lengths, p-cores, abacaus stuff..): they have a natural action on lablellings by S_n hence they are an S_n permutation module, they have an ordering respected by the action and induction and it all seems quite clear why to me, and yes Fulton explains all of this and far much more geometry besides (flag varieties and schubert calculus).