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The standard representation of $S_{n}$ has character equal to the number of fixed points minus $1$

Example

For $n \geq 1$ , let $S_{n}$ act on ${1, \dots, n}$ by permutation and let $C^{n}$ be the permutation representation. The standard representation is the subrepresentation

$V_{s t d} : = {(x_{1}, \dots, x_{n}) \in C^{n} : \sum_{i} x_{i} = 0},$

and its character is $χ_{s t d} (σ) = fix (σ) - 1$ , where $fix (σ)$ is the number of fixed points of $σ$ .

Facts & Assumptions

Given: An integer $n \geq 1$ and a permutation $σ \in S_{n}$ acting on ${1, \dots, n}$ .

[F1]

The character of a permutation representation is the number of fixed points of the action (The character of a permutation representation counts fixed points).

[F2]

Characters add on direct sums (Characters add on direct sums, multiply on tensor products, and conjugate on duals).

[A1]

The permutation representation decomposes as $C^{n} = L \oplus V_{s t d}$ , where $L = {(x, \dots, x) : x \in C}$ is a copy of the trivial representation and $V_{s t d}$ is its stated complement.

Verification

technique · direct

1.1

The line $L$ of constant vectors is fixed pointwise by every permutation, so it is a trivial subrepresentation, and the sum map $(x_{1}, \dots, x_{n}) \mapsto \sum_{i} x_{i}$ is preserved by every permutation, so its kernel $V_{s t d}$ is a subrepresentation. Every vector decomposes uniquely as a constant vector plus a vector of sum zero, which is [A1].

A1given

1.2

By [F2] applied to [A1], the permutation character $χ_{C^{n}}$ satisfies $χ_{C^{n}} = χ_{t r i v} + χ_{s t d}$ . The trivial character is the constant $1$ , so $χ_{s t d} (σ) = χ_{C^{n}} (σ) - 1$ .

F2A1given

2.1

By [F1], $χ_{C^{n}} (σ) = fix (σ)$ , the number of fixed points. Combining with step 1.2 gives $χ_{s t d} (σ) = fix (σ) - 1$ .

F1step 1.2algebra∎

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof et al., Introduction to Representation Theory, Section 3.7 (standard reference, not scraped)

The standard representation of SnSn​ has character equal to the number of fixed points minus 11