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The sign representation of $S_{n}$ and the restriction ${Res}_{H}^{G} (V)$ of a representation to a subgroup

Definition

Let $k$ be a field.

For the symmetric group $S_{n}$ (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition), the sign representation over $k$ is the one-dimensional representation on $k$ in which $σ \cdot a : = sgn (σ) a .$ This is a representation because $sgn : S_{n} \to {\pm 1}$ is a group homomorphism (The sign is a homomorphism $S_{n} \to {+ 1, - 1}$ , surjective exactly when $n \geq 2$ ).

Now let $ρ : G \to GL (V)$ be a representation of a group $G$ over $k$ , and let $H \leq G$ be a subgroup (Subgroup). The restriction of $ρ$ to $H$ is the representation ${Res}_{H}^{G} (V) : = ρ ∣_{H} : H ⟶ GL (V), h ⟼ ρ (h) .$ It has the same underlying vector space $V$ and only forgets the action of the elements outside $H$ .

Remarks

In characteristic $2$ , the sign representation coincides with the trivial representation because $- 1 = 1$ in the field.
Restriction changes the acting group, not the underlying vector space.

Depends on

Used by

Outer Pieri rules for a trivial or sign factor Corollary
The complex Specht restriction branching rule Corollary
p-regular and p-restricted labels differ Counterexample
An extension of a normal subgroup representation Definition
The restriction coproduct on the graded symmetric-group character ring Definition
The skew multiplicity module K^λ/μ over ℂ Definition
All three Specht modules of S₃ Example
Decomposition matrices of S3 at p=2 and p=3 Example
S₃ has three irreducible complex characters of degrees 1, 1, and 2 Example
S₃ is split over the rationals Example
The character table of S₄ and the normal subgroups it reveals Example
The regular representation of C₂ over a field of characteristic not 2 is the direct sum of the trivial and sign representations Example
The row and column Specht modules Example
Conjugate Specht modules are sign-twisted duals over every field Lemma
p-regular and p-restricted labels under transpose and sign Proposition
Sign twist corresponds to the omega involution Proposition
Specht restriction has a removable-corner filtration over every field Theorem

Dependency tree · two levels

19 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

Peter Webb, A Course in Finite Group Representation Theory, Chapter 4 Section 4.3 (standard reference, not scraped)

The sign representation of SnSn​ and the restriction Res⁡HG(V)ResHG​(V) of a representation to a subgroup