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The seven conjugacy classes of and their centralizer and class sizes
Example
The conjugacy data for are
| cycle type | centralizer size | class size | parity | splits in ? |
|---|---|---|---|---|
| even | no | |||
| odd | -- | |||
| even | no | |||
| even | no | |||
| odd | -- | |||
| odd | -- | |||
| even | yes |
Facts & Assumptions
Given: The symmetric group .
A cycle type has centralizer size (If has cycles of length , then ).
-classes are indexed by the tuples with (The conjugacy classes of are indexed by the tuples with ), and over those tuples, the summand indexed by being the size of the corresponding class (The class equation of is ).
A -cycle has sign , and when fixed points are included as -cycles (A -cycle has sign , and when fixed points are counted as cycles).
For and , the -class of splits into two -classes of equal size exactly when all cycle lengths in its decomposition, including -cycles for fixed points, are odd and no two are equal (For , an -class of an even permutation splits in exactly when all cycle lengths, including -cycles, are odd and distinct).
Verification
The seven partitions of give the seven rows. Applying [F1] gives the centralizer column, and [F2] gives the class-size column.
The sizes sum to , verifying the class equation.
Formula [F3] gives the parity column. Among the even rows, [F4] applies only to the single cycle of length , giving the final column.
Depends on
- The conjugacy classes of $S_n$ are indexed by the tuples $(c_1,\ldots,c_n)$ with $\sum k c_k=n$
- The class equation of $S_n$ is $n!=\sum_{\sum k c_k=n} n!/\prod_k k^{c_k}c_k!$
- If $\sigma\in S_n$ has $c_k$ cycles of length $k$, then $|C_{S_n}(\sigma)|=\prod_{k=1}^n k^{c_k}c_k!$
- A $k$-cycle has sign $(-1)^{k-1}$, and $\operatorname{sgn}(\sigma)=(-1)^{n-c(\sigma)}$ when fixed points are counted as cycles
- For $n\ge2$, an $S_n$-class of an even permutation splits in $A_n$ exactly when all cycle lengths, including $1$-cycles, are odd and distinct
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 61 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- K. Conrad, Conjugacy Classes (standard reference, not scraped)