9
$\begingroup$

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

A learning roadmap for Representation Theory

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.

Improve this question
$\endgroup$

7 Answers 7

Reset to default
15
$\begingroup$

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

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ 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. $\endgroup$
    – Jim Humphreys
    Commented Apr 19, 2011 at 12:57
13
$\begingroup$

"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").

Improve this answer
$\endgroup$
3
  • 3
    $\begingroup$ 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. $\endgroup$
    – darij grinberg
    Commented Apr 19, 2011 at 16:54
  • $\begingroup$ 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. $\endgroup$
    – Roland Bacher
    Commented Apr 20, 2011 at 6:31
  • $\begingroup$ I heart the Okounkov-Vershik approach! $\endgroup$
    – Noah Snyder
    Commented Apr 20, 2011 at 16:40
6
$\begingroup$

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.

Improve this answer
$\endgroup$
4
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • 5
    $\begingroup$ That being typed up is among my first memories. $\endgroup$
    – Allen Knutson
    Commented Apr 22, 2011 at 2:55
4
$\begingroup$

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.

Improve this answer
$\endgroup$
2
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ I second this choice. $\endgroup$
    – Sammy Black
    Commented Apr 19, 2011 at 21:38
1
$\begingroup$

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.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .