We want to come up with some nice sets for our symmetric group to act on. Our first step in this direction is to define a “Young tableau”.
If 
is a partition of 
, we define a Young tableau of shape 
to be an array of numbers. We start with the Ferrers diagram of the partition , and we replace the dots with the numbers 
to in any order. Clearly, there are 
Young tableaux of shape if .
For example, if 
, the Ferrers diagram is
We see that 
, and so there are 
Young tableaux of shape 
. They are
We write 
for the entry in the 
place. For example, the last tableau above has 
, 
, and 
.
We also call a Young tableau 
of shape a “-tableau”, and we write 
. We can write a generic -tableau as 
.
December 9, 2010
Posted by John Armstrong |
Algebra, Group theory, Representation Theory, Representations of Symmetric Groups |
13 Comments
