One concept I’d like to introduce is that of the “cycle type” of a permutation. This simply counts the number of cycles of various length in a permutation. For example — using the sample permutations from last time — 
has two cycles of length two, while 
has one cycle of length three and one “fixed point” — a cycle of length one. No cycle in a permutation in 
can be longer than 
, so we only need to count up to . We collect the information in a cycle type into a tuple of the form 
. The cycle type of 
is 
, and that of 
is 
.
It should be clear that the sum of the cycle lengths is . In a formula:
That is, the cycle type breaks up or “partitions” into chunks whose total size adds up to . In general, a partition 
of is a sequence of numbers 
in nonincreasing order, whose sum is . Thus the cycle type of gives us the partition 
, while the cycle type of gives us the partition 
.
One more example, from the beginning: the two-line notation
describing a permutation in 
has the cycle notation 
. Its cycle type is 
, which corresponds to the partition 
.
September 9, 2010
Posted by John Armstrong |
Algebra, Group theory, Representation Theory, Representations of Symmetric Groups |
1 Comment
