Introduction
What’s the motivation for introducing linear and unitary groups?
The study of group structures can be complicated so it can be useful to consider other groups whose algebraic structures are almost the same (i.e. nearly isomorphic), but with well-known elements and group multiplication (e.g. cyclic or permutation groups). Here the linear and unitary groups come in.
Linear and unitary groups
What’s the (general) linear group?
GL(V) = {A : V → V | det A ≠ 0}, over V (a linear space with a field of scalars) consist of all invertible linear operators on V, with product the composition (multiplication) of operators.
- linear group: a subgroup of GL(V)
- a subgroup of a linear group is itself linear
- dimension of a linear group = dimension of the linear space (cardinality of any basis)
- special linear group: det = 1 (unimodular operators)
Linear and unitary groups
Why are linear groups interesting?
Linear groups are more amenable to study because they allow the use of:
- linear algebra methods (e.g.: spectral decomposition)
- constructions specific to linear groups (e.g.: tensor products)
- special algorithms applicable only to collections of linear operators
Linear groups are rather the exception than the rule, but some special classes of groups (like finite, compact Lie, etc.) may be shown to have all members isomorphic to some linear group.
Linear and unitary groups
What is a complex inner product space? What’s a Hilbert space?
A complex linear space endowed with a map ⟨, ⟩ : H×H → C that is:
- linear in its first argument: ⟨λa+μb, c⟩ = λ⟨a, c⟩ + μ⟨b, c⟩
- conjugate symmetric: ⟨b, a⟩ = overline(⟨a, b⟩)
- positive definite: ⟨a, a⟩ > 0, for a ≠ 0
Hilbert space: a complex inner product space that is complete (as a metric space) with respect to the norm topology induced by the metric d(a, b) = sqrt(⟨a − b, a − b⟩).
Linear and unitary groups
What are (anti)unitary operators? What’s the unitary group?
Unitary operator: a linear op. U: H → H on the Hilbert space H, preserving the inner product
- ⟨Ux, Uy⟩ = ⟨x, y⟩
- they’re invertible
Antiunitary operator: an antilinear op. A: H → H on the Hilbert space H
- A (αx+βy) = conj(α)Ax + conj(β)Ay
- ⟨Ax, Ay⟩ = ⟨x, y⟩ = ⟨y, x⟩
- inverse is antiunitary, product is unitary
Unitary group: any subgroup of U(H) (or U(n)), formed by the products of unitary operators
- the collection házikó(U(H)) of all unitary or antiunitary operators is itself a group, having U(H) as a subgroup
Linear and unitary groups
Why are the unitary groups important in applications?
Unitary groups are easier to study because
- to each linear subspace of a Hilbert space one can associate its so-called orthogonal complement,
- the metric structure of a Hilbert space allows to single out especially useful bases (so-called orthonormal bases B = {e1, . . . , en} that satisfy ⟨e(i), e(j)⟩ = δ(ij)),
- any unitary operator can be diagonalized, and its eigenvalues lie on the complex unit circle U(1) = {z ∈ C | zz = 1}.
Linear and unitary groups
When is a linear group unitarizable?
A group G < GL(V ) is unitarizable, if V may be endowed with an inner product making it a Hilbert space H in such a way that G < U(H).
Linear and unitary groups
Why are unitary groups important specifically in physics?
Quantum systems:
Wigner’s theorem: the symmetries of a quantum system correspond to (anti-)unitary operators commuting with its Hamiltonian
- Hamiltonian = generator of time translations
- antiunitary symmetries —» time reversal symmetry
Gauge symmetries:
Gauge symmetries of fundamental interactions are described by (special) unitary groups in the Standard Model.
Matrix representations
What do the following mean: numerical representations, defining representations, matrix group?
Defining representation: if V has finite dimension n, then for any choice of basis B = {e1, . . . , en} there exist a (non-canonical) isomorphism ΓB : GL(V ) → GLn(C) that assigns cursive(A) to A, where A is the numerical representation
Matrix group: the image ΓB(G) of a lin. group
Matrix representations
What’s the difference between matrices and linear operators?
With respect to different bases, one and the same operator could be represented by different matrices, and one and the same matrix may represent different operators.
- so they are not the same
- ΓB′ (A) = C^(−1)ΓB(A)C, where B’ is another basis and C is the matrix of the basis change
Matrix representations
What’s a linear subspace?
A subset W ⊆ V of a linear space V (with field of scalars F) is a linear subspace if it contains all linear combinations of its elements.
- a linear subspace is nontrivial if it is different from the zero subspace and the whole space
- the translates x+W = {x+y | y ∈ W} of a linear subspace (where x ∈ V) form a new linear space, the factor space V/W with insert képletek
- any basis B(W) of a linear subspace W ⊆ V can be completed (in many different ways) to a basis B(V) ⊇ B(W) of the whole space
Matrix representations
What’s an invariant subspace?
A linear subspace W ⊆ V is an invariant subspace of the linear group G < GL(V ) if all group elements map it onto itself, meaning gx ∈ W for all g ∈ G and x ∈ W.
- so we don’t leave the subspace
- reduction of G: the linear group G(W) = {gW | g ∈ G} < GL(W) made up of the restrictions gW : W → W, x → gx
- factor of G: the linear group G/W = {g/W | g ∈ G} < GL(V /W) made up of the factored operators g/W: V /W → V /W, x + W → gx + W
Matrix representations
When are groups reducable and irreducable? What is Schur’s lemma for groups?
A linear group is reducible or irreducible according to whether it has a nontrivial invariant subspace.
Schur’s lemma: an operator that commutes with all elements of an
irreducible linear group is a scalar multiple of the identity operator
Linear representations
What’s a linear representation?
A linear representation of the group G over the linear space V is a homomorphism D : G → GL(V) into the general linear group over V .
- goal: given an abstract group, find a linear group isomorphic to it, but there’s no solution for a generic group, so we have to find suitable homomorphic images
- the rep. is faithful if its kernel is trivial
- the image of a representation is always a linear group that is isomorphic to G precisely when the representation is faithful
- complex representations: representations over finite dimensional complex linear spaces
Linear representations
What’s a matrix representation?
A linear representation D : G → GL(V ) over a linear space of dimension n determines, for any choice of a basis of the linear space V, a degree n matrix representation DB : G → GLn(F) via the rule DB = ΓB ◦D, where ΓB : GL(V ) → GLn(F) is the isomorphism associated to the basis B.
Linear representations
What are important examples of representations?
- Trivial rep.: for any group G and lin. space V, 1V : G → GL(V ), g → idV
- Defining rep.: for a linear group G < GL(V), the inclusion map, DG : G → GL(V ), g → g
- Permutation rep.: for any subgroup G < Sym(X)
- Unitary rep.: for group G on a Hilbert space H, a homomorphism U : G → U(H), so a linear representation all of whose representation operators are unitary (⟨U (g)x, U (g)y⟩ = ⟨x, y⟩)
- Unitarizable rep.: D: G → GL(V) is a rep. for which there exists a positive definite scalar product on V (making it a Hilbert-space) for which all representation operators are unitary
- all representations of a finite or compact group are unitarizable
Equivalence and reducibility
When are two representations equivalent?
The representations D1 : G → GL(V1) and D2 : G → GL(V2) are (linearly) equivalent, denoted D1 ∼= D2, if there exists an invertible linear map (intertwiner) A : V1 → V2 such that D2(g)A = AD1(g) for all g ∈ G.
- properties of equivalence: reflexive, symmetric, transitive
- equivalent representations are practically the same, e.g. the representation matrices of group elements coincide
Equivalence and reducibility
When are representations reducable and irreducable? What is Schur’s lemma for representations?
A representation D : G → GL(V ) is called reducible if its image is a reducible linear group (i.e. D(G) < GL(V ) has a nontrivial invariant subspace W < V ), otherwise it is called irreducible.
- 1D reps are always irreducible
Schur’s lemma: any operator that commutes with all representation operators of an irreducible representation is a multiple of the identity
- application: degeneracy of energy levels can be related to symmetries
Direct sum of representations
How to define the direct sum of representations?
Given representations D1: G → GL(V1) and D2: G → GL(V2), the map D1 ⊕D2 : G → GL(V1 ⊕V2), g → D1(g) ⊕ D2(g) is a new representation, the direct sum of D1 and D2, whose equivalence class is completely determined by the classes of D1 and D2.
- V1 ⊕V2 is the direct sum of linear spaces, whose elements are ordered pairs (x1, x2) with x1 ∈ V1 and x2 ∈ V2
- properties: commutativity, associativity (up to equivalence)
Direct sum of representations
When is a representation completely reducible? What is an irreducible decomposition?
A representation is completely reducible if it can be decomposed into a direct sum of irreducible representations.
Completely reducible representations have an irreducible decomposition:
(dir.sum: i∈Irr(G)) n(i) i
into a direct sum of irreducibles, where n(i) ∈ Z+ is the multiplicity of the irreducible i ∈ Irr(G).
- so there’s a one-to-one correspondence between completely reducible reps and maps from Irr(G) into Z+
Direct sum of representations
What’s Maschke’s theorem? What’s the Peter-Weyl theorem?
All complex representations of a finite group are completely reducible.
- Peter-Weyl: a similar result, holds for the representations of compact topological groups
- more generally, all unitary representations are completely reducible
Tensor products and the fusion ring
What are bilinear functionals and what is the dyadic product?
Bilinear functional: on the linear spaces V1 and V2 (with common field of scalars F) is a map b: V1×V2 → F that is linear in each of its arguments
- b(αx1 + βy1, x2) = αb(x1, x2) + βb(y1, x2) and b(x1, αx2 + βy2) = αb(x1, x2) + βb(x1, y2)
- the set B(V1, V2) of bilinear functionals is a linear space with the pointlike operations b1 + b2: V1 ×V2 → F and αb: V1 ×V2 → F
Dyadic product: v1⊗v2 of v1 ∈ V1 and v2 ∈ V2 is the linear functional v1⊗v2 : B(V1, V2) → F, b → b(v1, v2)
- in other words: the evaluation of bilinear functionals at the given arguments
Tensor products and the fusion ring
What is the tensor product of:
- linear spaces,
- linear operators?
Linear spaces: the dual of the space of bilinear functions spanned by the dyadic products, V1⊗V2 = B(V1, V2)∨
Linear operators: A1⊗A2: V1⊗V2 → W1⊗W2 that maps each dyadic product v1⊗v2 into A1v1⊗A2v2 for A1: V1 → W1 and A2: V2 → W
- the matrix of the tensor product with respect to the product basis {e⊗f | e ∈ B1, f ∈ B2}, is the Kroenecker product of the matrices ΓB1(A1) and ΓB2(A2)
Tensor products and the fusion ring
What is the tensor product of representations?
D1⊗D2 : G → GL(V1⊗V2), g → D1(g)⊗D2(g) for D1: G → GL(V1) and D2: G → GL(V2)
- compatible with linear equivalence, i.e. the class of a product is determined by the classes of its factors
- commutative, associative, distributive with respect to sums
- the identity rep. is an identity element for the tensor product
