Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 September 2013

Appendix A

A2. Partitions and tableaux


A partition is a sequence λ=(λ1,,λn) of integers such that λ1λn0. It is conventional to identify a partition with its Ferrers diagram which has λi boxes in the ith row. For example the partition λ=(55422211) has Ferrers diagram λ=(55422211) We number the rows and columns of the Ferrers diagram as is conventionally done for matrices. If x is a box in λ then the content and the hook length of x are respectively given by c(x)=j-i, ifxis in position (i,j)λ, and hx=λi-i+λj-j+1, whereλj is the length of thejth column ofλ. 0 1 2 3 4 -1 0 1 2 3 -2 -1 0 1 -3 -2 -4 -3 -5 -4 -6 -7 12 9 5 4 2 11 8 4 3 1 9 6 2 1 6 3 5 2 4 1 2 1 Contents of the boxes Hook lengths of the boxes

If μ and λ are partitions such that the Ferrers diagram of μ is contained in the Ferrers diagram of λ then we write μλ and we denote the difference of the Ferrers diagrams by λ/μ. We refer to λ/μ as a shape or, more specifically, a skew shape. λ/μ=(55422211)/(32211)


Suppose that λ has k boxes. A standard tableau of shape λ is a filling of the Ferrers diagram of λ with 1,2,,k such that the rows and columns are increasing from left to right and from top to bottom respectively. 1 2 5 9 13 3 6 10 14 16 4 8 15 17 7 12 11 20 18 21 19 22 Let λ/μ be a shape. A column strict tableau of shape λ/μ filled with 1,2,,n is a filling of the Ferrers diagram of λ/μ with elements of the set {1,2,,n} such that the rows are weakly increasing from left to right and the columns are strictly increasing from top to bottom. The weight of a column strict tableau T is the sequence of positive integers ν=(ν1,,νn), where νi is the number of i’s in T. 1 1 1 2 3 2 2 4 4 5 3 3 5 5 6 7 7 8 8 9 10 11 Shapeλ=(55422211) Weightν=(33323122111) The word of a column strict tableau T is the sequence w=w1w2wp obtained by reading the entries of T from right to left in successive rows, starting with the top row. A word w=w1wp is a lattice permutation if for each 1rp and each 1in-1 the number of occurrences of the symbol i in w1wr is not less than the number of occurences of i+1 in w1wr. 1 1 1 2 2 3 4 3 4 5 6 7 8 1 1 1 2 2 3 3 4 5 5 6 7 8 w=1122143346578 w=1122133456578 Not a lattice permutation Lattice permutation A border strip is a skew shape λ/μ which is

(a) connected (two boxes are connected if they share an edge), and
(b) does not contain a 2×2 block of boxes.
The weight of a border strip λ/μ is given by wt(λ/μ)= (-1)r(λ/μ)-1, where r(λ/μ) is the number of rows in λ/μ. λ/μ = (86333)/(5222) wt(λ/μ) = (-1)5-1
Let λ and μ=(μ1,,μ) be partitions of n. A μ-border strip tableau of shape λ is a sequence of partitions T= ( = λ(0) λ(1) λ(-1) λ()=λ ) such that, for each 1i,
(a) λ(i)/λ(i-1) is a border strip, and
(b) |λ(i)/λ(i-1)|=μi.

The weight of a μ-border strip tableau T of shape λ is wt(T)= i=1-1 wt(λ(i)/λ(i-1)). (A2.1)

Theorem A2.2. (Murnaghan-Nakayama rule) Let λ and μ be partitions of n and let χλ(μ) denote the irreducible character of the symmetric group Sn indexed by λ evaluated at a permutation of cycle type μ. Then χλ(μ)= Twt(T), where the sum is over all μ-border strip tableaux T of shape λ and wt(T) is as given in (A2.1).


All of the above facts can be found in [Mac1995] Chapt. I. The proof of theorem (A2.2) is given in [Mac1995] Ch. I §7, Ex. 5.

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

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