# Representation theory of $S_n$

I need to understand the representation theory of $S_n$ (symmetric group on $n$ letters) and so could someone suggest a good reference for this. There is a similar question on mathoverflow here

Most of the responses to the above question give references for representation theory of Lie groups. Also the usual reference Fulton and Harris has too many exercises (on which I don't want to spend too much time ) and I find it difficult to read.

Another reference which was suggested was Flag varieties by Lakshmibai and Brown. This seems to be a good reference, but are there any other references.

EDIT: By mistake I did not notice something in the above mentioned book and so some of my remarks are being edited. Sorry.

-

## 7 Answers

"The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions" by Bruce Sagan might be a good place to start.

-
There's no need to study representations and combinatorics of symmetric groups in any wider representation context (as in Goldschmidt, Fulton-Harris, etc.), though that's often where serious applications occur. Sagan's modern textbook tends to follow the approach of the influential lecture notes by Gordon James, emphasizing characteristic-free formulations. This facilitates passage to prime characteristic where much recent research has been focused. – Jim Humphreys Apr 19 '11 at 12:57
Thanks. It looks interesting. I will take a look at it. – Rex Apr 20 '11 at 2:39

"Group characters, symmetric functions and the Hecke algebra" by D. M. Goldschmidt is also very nice.

And there is of course also the classical "Symmetric functions and Hall polynomials" by Macdonald.

In a second time you can also have a look at the Okounkov-Vershik approach (perhaps by reading the original paper "A new approach to representation theory of symmetric groups").

-
There is also "Representation Theory of the Symmetric Groups - The Okounkov–Vershik Approach, Character Formulas, and Partition Algebras" by Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli. I have no real experience with this book, but I wouldn't be surprised if it is way more readable than the Okounkov-Vershik original paper. – darij grinberg Apr 19 '11 at 16:54
The Okounkov-Vershik paper is fairly elementary and not so difficult to read. However, I do not know the book by Ceccherini-Silberstein, Scarabotti and Tolli. I will have a look at it. – Roland Bacher Apr 20 '11 at 6:31
I heart the Okounkov-Vershik approach! – Noah Snyder Apr 20 '11 at 16:40

In my opinion some good references are "Representation theory of the symmetric group" by "James G, Kerber A." and "The representation theory of the symmetric group" (Lecture notes in mathematics) by G.D. James.

-

I recommend $\lambda$-Rings and the Representation Theory of the Symmetric Group by Donald Knutson. It helped me a lot. It's #308 in the Springer Lecture Notes series.

-
That being typed up is among my first memories. – Allen Knutson Apr 22 '11 at 2:55

If you like combinatorics, you may enjoy learning about the representations of $S_n$ by reading Chapter 7 of Stanley's Enumerative Combinatorics, Volume 2.

-

I guess I should plug Sasha Kleshchev's book "Linear and Projective Representations of Symmetric Groups". Chapter 1 reconstructs the representation theory of symmetric groups from the Jucys-Murphy elements. The remainder of the book puts this representation theory into the larger context of cyclotomic Hecke algebras.

I would recommend paying close attention to the intertwining operators that are introduced in Chapter 3. A good exercise would be to use these operators to recover the results of Arun Ram's papers "Calibrated Representations of Affine Hecke Algebras" and "Skew Shape Representation are Irreducible". Once you do this, you will understand Young's semi-normal form, and how to construct irreducible representations of $S_n$ from semi-standard Young tableaux.

-
I second this choice. – Sammy Black Apr 19 '11 at 21:38

Note that both the James and James/Kerber classic books are back in print and available from Amazon. The new book "Representation Theory of the Symmetric Groups" by Ceccherini-Silberstein et al is quite nice.

-