T1: Lectures 1-5 Flashcards
Definition of a group
A triple including a set, binary operation and identity element.
Give two additional criteria for a group
The binary operation is associative and for each element there exists an inverse.
Define the symmetric group S_n
Characterises the permutations of n elements.
Order n!
Give the presentation of S_3
Where s=(1 2), t=(2 3) we have:
<s,t|s^2=t^2=e, sts=tst>
Define the dihedral group D_n
The set of symmetries of a regular n-gon inscribed on the unit circle with a point at (1,0)
Size 2n
How many rotation and reflections are in D_n (and hence give the size/order of the group)
n rotations and n reflections (hence, 2n).
Define the quaternion group Q_8
The set of elements +_{1,i,j,k} with i^2=j^2=k^2=ijk=-1
Define GL_n(V)
The set of n x n matrices with non zero determinant that characterise a linear map from V to V.
The function φ mapping G to H two groups (G, ⋅) and (H, *) is a homomorphism if..?
φ(g_1)*φ(g_2)
= φ(g_1 ⋅ g_2)
What are the two key ways to verify a homomorphism?
Check the homomorphism relation, or verify that the map preserves the presentation of the original group.
Define a representation of a group (π,V)
A pair (π,V) where v is a vector space, and the map π: G to GL_n(V) is a group homomorphism
π(g)π(h)=π(gh) for all g,h in G.
What is the permutation representation of the symmetric group S_n?
(column vector)
(π, C^n) where we have an n dimensional vector with entries holding:
π(σ)(z_n)=(z_(σ^(-1)(n))
What is the permutation representation of the symmetric group S_n?
(unit vector)
(π, C^n) where we have an n term sum with entries holding:
π(σ)(x_1e_1+ … +x_ne_n)
= x_1e_σ(1)+ … + x_ne_σ(n).
Define the trivial representation of a group G
(π,V) where π: g to Id
for all g in G
What is the dimension of a representation?
The dimension of the underlying vector space.
What is the defining rep for D_n
The pair of matrices ρ(r) and ρ(s)
-sin top right
-1 top left
How can we show that a set of matrices represent a group
The same way as showing a homomorphism: check the homomorphism condition or verify the presentation.
How does a subspace form a subrepresentation?
Let (π,V) be a rep of G. A subspace W⊆V forms a subrepresentation (π,W) if for all π(g)w the result is inside W.
Define an irreducible representation
A rep (π,V) is irreducible if it has no non-trivial subrepresentations: (π,0) and (π,V).
When finding subgroups, what is the condition on the order of elements and subgroups?
The order of any element and any subgroup must divide the order of the group.
How can we determine a conjugacy class?
conjugate all elements of the group between the generators and match up corresponding (or cyclic) results.
What is a condition on a conjugacy class?
All elements have the same order.
What is the order of a group?
The number of elements
What is the order of an element in a group?
The power you must raise it to to return the identity.
Consider the permutation rep of S_n and give the two subspaces for the subreps
W = complex number * n-dim column vector of ones
W^⊥ = n dim column vector st sum of elements = 0
How do the subspaces in the sub reps of S_3 relate to one another and the whole vector space of S_3?
W and W^⊥ are orthogonal and their direct sum = V
How can we prove that a representation is irreducible.
Consider an irreducible subrep of the rep in question and show that the vector spaces must be equal.
Complete the remark: “Every representation of a finite group spanning a complex vector space…
…can be decomposed into a direct sum of irreducble reps.
Consider two reps of a group G: (π_1,V) and (π_2,W). The linear map T:V to W is a hom if:
Tπ_1(g) = π_2(g)T for all g in G
(alt T(π_1(g)v) = π_2(g)T(v) )
Hom_G (V,W) denotes..
The vector space of all G homomorphisms from V to W.
Give the condition for a G-hom to be an isomorphism
The linear map T (and hence T^-1) must be invertible.
Given T is a G-hom between (π_1,V) and (π_2,W), define a subrep for each of the latter.
(π_1, ker(T)) and (π_2, Im(T)) are subreps of (π_1,V) and (π_2,W) respectively.
Define Ker(T)
The set of vectors for which Tv = 0
Define Im(T)
The set of vectors w for which w = Tv
State Schur’s Lemma (part ii)
If (π,V) is a finite-dimensional irreducible representation of group G, then Hom_G (V)=CId
Define Abelian Group
A group for which the group operation is commutative.
State the first isomorphism theorem
For T: V to U
The map taking quotient space V/Ker(T) to U is isomorphic to taking V/Ker(T) to Im(T)
Consider first iso theorem. How do the dim kernel and image relate to the dim of V
dim ker + dim im = dimV
What are the kernel and image of a Hom f: G to G’ between groups
ker(f) = g in G, f(g) = e
im(f) = g in g, f(g)
State Schur’s Lemma (part i)
For T in HomG(V,W) between two irreducible finite complex reps, either T is an iso, or T = 0
State Schur’s Lemma (part iii)
dimHomG(V,W) for a irr rep is 1 if V and W are isomorphic or 0 else.
What is special about Abelian groups?
Every finite dim irr rep is one-dimensional.
