MATH 817 Notes
JD Nir
[email protected]
www.math.unl.edu/∼jnir2/817.html
September 30, 2015
Recall:
Sn acts on R[x1 , x2 , . . . , xn ]
∗ σ · (f g) = (σ · f )(σ · g)
∗ σ · (rf ) = rσ · f
Note: If σ · f = f ∀σ ∈ Sn , f is a symmetric polynomial:
e.g

• 1

X



• x1 + x2 + . . . + xn =
xi 




1≤i≤n

X



•
xi xj

1≤i<j≤n
elementary symmetric polynomials
X


•
xi xj x`




1≤i<j<`≤n



..



.


• x1 · · · xn
X
i < jxi xj
x21 + . . . + x2n = (x1 + . . . + xn )2 − 2
Y
Define ∆ =
(xi − xj )
1≤i<j≤n
Lemma σ · ∆ = ±∆ ∀σ ∈ Sn
Y
Pf σ · ∆ =
(xσ(i) − xσ(j) )
i<j
xσ(i) −xσ(j) is either a factor of ∆ or the negative of a factor. Moreover, each term of ∆ corresponds
to only one term of σ · ∆ or a negative of such a term.
∆
= ±1.
So,
σ·∆
Def If σ ∈ Sn ,
(
+1 if σ · ∆ = ∆
sgn(σ) :=
−1 if σ · ∆ = −∆
∆
= σ·∆
1
JD Nir
[email protected]
MATH 817
Prop: 1 sgn : Sn → {±1} is a group homomorphism.
2 It’s onto
3 If σ is a product of ` transpositions, then sgn(σ) = (−1)`
Pf 1
(σ ◦ τ ) · ∆ = σ · (τ · ∆)
= σ · (sgn(τ )∆)
= sgn(τ )(σ · ∆)
= sgn(τ ) sgn(σ)∆
= sgn(σ) sgn(τ )∆
(σ ◦ τ ) · ∆ = sgn(σ ◦ τ )∆
∴ sgn(σ ◦ τ ) = sgn(σ) sgn(τ ) X
If σ = (1 2),
σ · ∆ = (x2 − x1 )(x2 − x3 )(x2 − x4 ) . . . (x2 − xn )
· (x1 − x3 )(x1 − x4 ) . . . (x1 − xn )
Y
·
(xi − xj )
3≤i<j≤n
= −∆
∴ sgn(1 2) = −1
∴ 2 holds.
If (a b), a 6= b, is any transposition, then (a b) = σ(a b)σ −1 , some σ ∈ Sn (By HW)
∴ sgn((a b)) = sgn(σ) sgn((1 2)) sgn(σ)−1
= sgn((1 2)) = −1, since {±1} is abelian.
If σ = τ1 · · · τ` with τi = a transposition ∀i,
sgn(σ) =
`
Y
sgn(τi ) = (−1)`
i=1
j
P
Cor: If σ is a product of cycles of length `1 , `2 , . . . , `j then sgn(σ) = (−1)i=1
`i −1
=
Q
((−1)`i −1 )
Pf Since sgn is a homomorphism, it suffices to show sgn(τ ) = (−1)`−1 if τ = (a1 a2 · · · a` ). But
τ = (a1 a2 )(a2 a3 ) · · · (a`−1 a` )
⇒ sgn(τ ) = (−1)`−1 .
Note sgn(σ) = 1 ⇔ in the unique disjoint cycle expression for σ, there are an even number of cycles
of even length.
2
MATH 817
JD Nir
[email protected]
Group Actions, Again
G = group, X = set, assume G acts on X (⇔ ∃ homomorphism ρ : G → Perm(X))
Def Write x ∼ y, x, y ∈ X if y = g · x, some g ∈ G.
Fact ∼ is an equivalence relation
• For x ∈ X, the orbit of x (under the given action) is G · x = {g · x | g ∈ G}
I.e., the orbits are the equivalence classes.
• The action is transitive if there is only one orbit
• The stabilizer of x ∈ X is Gx := {g ∈ G | g · x = x}
Ex 3 G = group of rotational symmetries of
G acts on the X of all faces of this cube. If f is a face,
Gf = {g ∈ G | g fixes f }
#Gf = 4
Theorem[LOIS] “Length of an Orbit is the Index of the Stabilizer”
If G acts on a set of X and x ∈ X, then
G : Gx ] = #G · x
(Note: Gx ≤ G)
Pf: Let L = {left cosets of Gx in G} = {gGx | g ∈ G}
If gGx = g 0 Gx , then g = g 0 h, some h ∈ Gx and thus g · x = (g 0 h) · x = g 0 · (h · x) = g 0 · x.
It follows that α : L → G · x given by α(gGx ) = g · x is well-defined. It’s obviously onto. If
α(gGx ) = α(g 0 Gx ), then
g · x = g 0 · x ⇒ (g −1 g 0 ) · x = x ⇒ g −1 g 0 ∈ Gx ⇒ gGx = g 0 Gx .
∴ α is 1-1. 3