character table of $S_5$

Let’s use GAP to find the character table of $S_5$.

gap> G:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> tab:=CharacterTable(G);
CharacterTable( Sym( [ 1 .. 5 ] ) )
gap> Display(tab);
CT1

     2  3  2  3  1  1  2  .
     3  1  1  .  1  1  .  .
     5  1  .  .  .  .  .  1

       1a 2a 2b 3a 6a 4a 5a
    2P 1a 1a 1a 3a 3a 2b 5a
    3P 1a 2a 2b 1a 2a 4a 5a
    5P 1a 2a 2b 3a 6a 4a 1a

X.1     1 -1  1  1 -1 -1  1
X.2     4 -2  .  1  1  . -1
X.3     5 -1  1 -1 -1  1  .
X.4     6  . -2  .  .  .  1
X.5     5  1  1 -1  1 -1  .
X.6     4  2  .  1 -1  . -1
X.7     1  1  1  1  1  1  1
gap> 

Diaconis example – survey data

This data is taken from the paper (Diaconis 1989)

It describes 5,738 completed ballots rank-ordering 5 candidates.

View a rank-ordered ballot as an element of the symmetric group $S_5$; we want to study the frequency function $f$.

first ranking table

the regular representation

This diagram shows the decomposition of the regular representation into isotypic components.

Be careful: the notation Diaconis is using here does not match that used by GAP above. For example, the representation Diaconis writes as $V_3$ is the isotypic component determined by the irreducible representation labeled X.5 by GAP.

The second row reflects the decomposition of the frequency function $f$. Namely, write $$f=\sum _{i=1}^7{f}_i\text{with }{f}_i\in V_i.$$

The second row entries are the “sums of squares” $⟨f_i,f_i⟩$.

Remember that we can compute the $f_i$ using the idempotents in $\mathbb{C}[G]$.

For example,

$$f_1={\displaystyle \frac{1}{5!}}\sum _{\sigma \in S_5}\sigma .f$$

More generally, if ${\chi }_i$ denotes the character of the irreducible representation $L_i$ with $V_i=\mathbb{C}[G{]}_{(L_i)}$ then $$f_i={\displaystyle \frac{1}{5!}}\sum _{\sigma \in S_5}{\chi }_i({\sigma }^{-1})\sigma .f$$

Note that $⟨f_3,f_3⟩=459$ is relatively large (ignoring $⟨f_1,f_1⟩$ since $f_1$ is trivial).

normalizing the first-order data

THe $i,j$ entry in this table is the number of votes ranking candidate $i$ in the $j$-th position, minus the sample size over 5.

In particular, rows and columns sum to 0.

This normalization can also be achieved as follows:

Let $f_2$ be the projection on $V_2$, and consider the functions $$\sigma \mapsto {\delta }_{i,\sigma (j)}.$$

The $i,j$ entry of the preceding table is $⟨f_2,{\delta }_{i,\sigma (j)}⟩$

Interpretation in this last table:

Compute the projection $f_3$ of $f$ into the component $V_3$ of $M=\mathbb{C}[S_5]$.

Now, consider the easily understood functions $$\sigma \mapsto {\delta }_{\{i,i^{\prime }\},\{\sigma (j),\sigma (j^{\prime })\}}$$ in $\mathbb{C}[S_5]$ for distinct $i,i^{\prime }$ and distinct $j,j^{\prime }$.

The space of these functions is a 100 dimensional subspace of $W$ $\mathbb{C}[G]$.

The entries in the table are the inner products $$⟨f_3,{\delta }_{\{i,i^{\prime }\},\{\sigma (j),\sigma (j^{\prime })\}}⟩$$

Summary observations

The data were to elect a president for the American Psychological Association. Candidates 1 and 3 were clinicians while candidates 4 and 5 were academicians, two groups within the association with somewhat divergent perspectives.

In the second-order table, we see a preference for candidates 1 & 3 witnessed by the entry 376 corresponding to the entry for candidates $\{1,3\}$ and ranks $\{1,2\}$.

And we see a (slightly smaller) preference for candidates 4 and 5 witnessed by the entry 296 corresponding to the entry for candidates $\{4,5\}$ and ranks $\{1,2\}$.

Bibliography