Last update: 17 September 2013
A partition is a sequence of integers such that It is conventional to identify a partition with its Ferrers diagram which has boxes in the row. For example the partition has Ferrers diagram We number the rows and columns of the Ferrers diagram as is conventionally done for matrices. If is a box in then the content and the hook length of are respectively given by
If and are partitions such that the Ferrers diagram of is contained in the Ferrers diagram of then we write and we denote the difference of the Ferrers diagrams by We refer to as a shape or, more specifically, a skew shape.
Suppose that has boxes. A standard tableau of shape is a filling of the Ferrers diagram of with such that the rows and columns are increasing from left to right and from top to bottom respectively. Let be a shape. A column strict tableau of shape filled with is a filling of the Ferrers diagram of with elements of the set 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 is the sequence of positive integers where is the number of in The word of a column strict tableau is the sequence obtained by reading the entries of from right to left in successive rows, starting with the top row. A word is a lattice permutation if for each and each the number of occurrences of the symbol in is not less than the number of occurences of in A border strip is a skew shape which is
|(a)||connected (two boxes are connected if they share an edge), and|
does not contain a block of boxes.
The weight of a border strip is given by where is the number of rows in
|(a)||is a border strip, and|
The weight of a strip tableau of shape is
Theorem A2.2. (Murnaghan-Nakayama rule) Let and be partitions of and let denote the irreducible character of the symmetric group indexed by evaluated at a permutation of cycle type Then where the sum is over all strip tableaux of shape and 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.
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..