(October 6, 2014) Generalities on representations of finite groups
Paul Garrett [email protected] http://www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/˜garrett/m/repns/notes 2014-15/03 generalities finite.pdf] 1. Subrepresentations, complete reducibility, unitarization 2. Dual/contragredient representations 3. Regular and biregular representations L2(G) 4. Schur’s lemma 5. Central characters of irreducibles 6. Tensor products of representations 7. Matrix coefficient functions 8. Schur orthogonality, inner product relations o 9. Representations of Cc (G), convolution o 10. G-representations versus Cc (G)-modules 11. Decomposition of biregular representation on L2(G) A representation of a finite group G on a finite-dimensional complex vector space V is a group homomorphism ρ : G → AutCV of G to the C-linear automorphisms of V . The vector space is completely specified by ρ, so, often, ρ denotes both the map from G and the vectorspace on which ρ makes G act. For further notational economy, instead of writing ρ(g)(v), we may write g · v or gv, when context permits. A G-morphism or G-homomorphism or G-map or G-intertwining operator
ϕ :(σ, V ) −→ (τ, W )
from one G-representation (σ, V ) to another (τ, W ) is, as expected, a vector-space map ϕ : V → W which commutes with or respects the action of G in the natural sense:
ϕ ◦ σ(g) = τ(g) ◦ ϕ (for all g ∈ G)
Vector-wise, using the lighter notation, the requirement is
ϕ(g · v) = g · ϕ(v) (for all v ∈ V )
The G-intertwining operators from (V, σ) to (W, τ) are denoted HomG(σ, τ).
1. Subrepresentations, complete reducibility, unitarization
A subrepresentation of a representation (ρ, V ) of G is a C-subspace W of V which is G-stable in the sense 0 0 that, for all g ∈ G and W ∈ W , ρ(g)(W ) ∈ W . With ρ : G → AutCW the restriction of ρ(G) to W ,(ρ ,W ) is a representation of G in its own right. The direct sum representation (σ, V ) ⊕ (τ, W ), or simply σ ⊕ τ, has representation space the direct sum V ⊕ W of the two Hilbert spaces, with the natural action
(σ ⊕ τ)(g)(v ⊕ w) = σ(g)v ⊕ τ(g)w
That is, more economically, g · (v ⊕ w) = gv ⊕ gw A representation (ρ, V ) of G is irreducible if there is no G-stable subspace of V other than {0} and V itself.
1 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
[1.0.1] Remark: A first important idealized goal of representation theory is to classify or parametrize irreducibles of given G in a useful way.
[1.0.2] Remark: For G abelian, the irreducibles are exactly given by the group homomorphisms χ : G → × C = AutCC, and are all one-dimensional. This follows from the spectral theorem for finite groups of mutually commuting operators. A representation (ρ, V ) of G is completely reducible when it is isomorphic to a direct sum of irreducible representations, that is, when there are irreducibles (ρi,Vi) such that M ρ ≈ ρi i
[1.0.3] Remark: For finite a abelian group G acting linearly on a finite-dimensional complex vector space, it is an exercise in linear algebra to prove that there is a basis for V of simultaneous eigenvectors for all operators coming from G. This is a decomposition into irreducibles.
[1.0.4] Remark: A second idealized goal of representation theory is to use a classification of irreducibles of L given G to usefully describe the irreducibles ρi appearing in a decomposition ρ = i ρi of naturally-occurring representations ρ of G, to analyze ρ.
The latter goal, of describing irreducible summands of larger representations, presumes that representations are reliably direct sums of irreducibles. This holds for finite-dimensional complex representations of finite groups:
[1.0.5] Theorem: Every finite-dimensional complex representation of a finite group is completely reducible.
Proof: The argument uses the notion of unitary representation, and unitarization of a given representation. A C-linear map T : V → V on a finite-dimensional complex Hilbert space V with inner product h, i is unitary when hT v, T wi = hv, wi (for all v, w ∈ V ) A product of two unitary operators is unitary, as is the inverse of a unitary operator, so the unitary operators on V form a group. A representation ρ of G on V is unitary when ρ(g) is a unitary operator on V for every g ∈ G.
[1.0.6] Claim: A representation ρ of G on a complex vector space V is unitarizable, meaning that there is a hermitian inner product h, i on V so that ρ is unitary.
Proof: Take any hermitian inner product h, io on V , and create the invariant h, i by averaging h, io over G:
1 X hv, wi = hgv, gwi #G g∈G
By design, this is G-invariant: for h ∈ G,
1 X 1 X 1 X hhv, hwi = hg(hv), g(hw)i = h(gh)v, (gh)wi = hgv, gwi #G o #G o #G o g∈G g∈G g∈G by replacing g by g−1 in the sum. The averaged inner product is still hermitian-linear: checking linearity in the first argument, for a ∈ C, 1 X 1 X hav + v0, wi = hg(av + v0), gwi = ahgv, gwi + hgv0, gwi #G o #G o o g∈G g∈G
2 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
1 X 1 X = a hgv, gwi + hgv0, gwi = ahv, wi + hv0, wi #G o #G o g∈G g∈G Conjugate-linearity in the second argument is similar, as is positive-definiteness. ///
Now we can prove the theorem by induction on dimC V . If there is no proper non-zero G-stable subspace of V , then V is irreducible, by definition. If there is a proper G-stable subspace W , then, by induction, W is L 0 a direct sum V = i Wi of irreducibles. If we can find a G-stable complementary subspace W to W , then 0 0 L 0 0 the induction hypothesis applies to W as well, so it is a direct sum W = j Wj of irreducibles Wj, and
0 M M 0 V = W ⊕ W = Wi ⊕ Wj i j expresses V as a direct sum of irreducibles. To find a complementary subspace to W , give V a G-invariant hermitian inner product h, i. We claim that the orthogonal complement W 0 of W is G-stable: using the G-invariance, that is, the unitariness of the action of G,
hgw0, wi = hg−1gw0, g−1wi = hw0, g−1i = 0 (for w0 ∈ W 0 and w ∈ W ) since g−1W ∈ W . This proves the G-stability of the orthogonal complement. Then the induction argument succeeds. ///
2. Dual/contragredient representations
∨ The dual space V of a complex vectorspace V is the complex vectorspace HomC(V, C) of C-valued linear functionals on V . The complex vectorspace structure is
(a · λ)(v) = a(λ(v)) (for λ ∈ V ∨, a ∈ C, and v ∈ V )
The contragredient or dual representation (ρ∨,V ∨) of G on V ∨ is
ρ∨(g)(λ)(v) = λ(ρ(g)−1v)
or, more economically, (g · λ)(v) = λ(g−1 · v) The possibly unexpected inverse exactly assures that
ρ∨(gh) = ρ∨(g) ρ∨(h) (for g, h ∈ G)
The complex-bilinear pairing V × V ∨ → C by v × λ → λ(v) is sometimes usefully denoted
λ(v) = hv, λi (complex bilinear)
When V has a hermitian inner product, the latter is often denoted h, i as well, inviting confusion. Further, for V with a hermitian inner product, the Riesz-Fischer theorem gives a complex-conjugate-linear isomorphism V → V ∨, by v −→ w → hw, vi (with hermitian inner product)
inviting further confusion. We will have reason to use hermitian inner products, such as that on L2(G), and to prove complete reducibility as earlier, but the complex bilinear pairing of V and V ∨ is equally important. Context should make clear which is meant.
[2.0.1] Claim: For ρ an irreducible of G, the dual/contragredient ρ∨ is irreducible.
3 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
Proof: For a G-subrepresentation X of ρ∨, the simultaneous kernel X0 of X in ρ is G-stable, because λ(g · v) = (g−1λ)(v) for all λ ∈ X. Since
0 ∨ dimC X + dimC X = dimC ρ = dimC ρ the simultaneous kernel X0 is a proper subspace of ρ if X is a proper subspace of ρ∨. ///
3. Regular and biregular representations
Let L2(G) be the square-integrable complex-valued functions on G using counting measure on G. The right regular representation R of G on L2(G) is defined by
2 R(g)f(h) = Rgf(h) = f(hg) (for g, h ∈ G and f ∈ L (G))
The left regular representation L is similarly defined, by
−1 2 L(g)f(h) = Lgf(h) = f(g h) (for g, h ∈ G and f ∈ L (G))
The inverse in the formula is necessary to have L(gg0) = L(g)L(g0) for non-abelian groups, as in the definition of contragredient representation. The biregular representation of G × G on L2(G) is
0 −1 0 ρbi(g × g )f(h) = f(g hg )
[3.0.1] Claim: The right regular, left regular, and biregular representations of G on L2(G) are unitary.
Proof: For the right regular representation, with f, F ∈ L2(G), and g ∈ G, using the definition and changing variables, X X hRgf, RgF i = f(xg) F (xg) = f(x) F (x) = hf, F i x∈G x∈G proving the unitariness. ///
4. Schur’s Lemma
[4.0.1] Theorem: For an irreducible ρ, V of G, a C-linear map T : V → V commuting with all operators ρ(g) for g ∈ G is a scalar.
Proof: The kernel of such T is G-stable:
T (g · v) = g · T v = g · 0 = 0 (for v ∈ ker T )
Since ρ is irreducible, ker T is either {0} or V itself. By dimension counting, T is either the 0-map or is a bijection. That is, the ring A of such T contains C, and is a division algebra. It is finite-dimensional over C, since V is finite-dimensional. Thus, any T ∈ A generates a finite algebraic extension of C. Since C is algebraically closed, T is scalar. ///
4 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
5. Central characters of irreducibles
[5.0.1] Corollary: (of Schur’s lemma) The center of G acts by scalars on an irreducible ρ, V of G. ///
[5.0.2] Remark: The restriction of ρ to the center of G is the central character of ρ, sometimes denoted ωρ.
6. Tensor products of representations
Given representations σ, V and τ, W of G and H, the (external) tensor product representation σ ⊗τ of G×H has representation space V ⊗C W , with action (σ ⊗ τ)(g × h)(v ⊗ w) = σ(g)(v) ⊗ τ(h)(w)
The (internal) tensor product of representations σ, V and τ, W of G is defined the same way, but restricting the group action to the diagonal copy G∆ of G inside G × G. That is,
(σ ⊗ τ)(g)(v ⊗ w) = σ(g)(v) ⊗ τ(g)(w)
[6.0.1] Remark: External and internal tensor products are distinguished by context. Some writers use a square tensor symbol for external tensor products, but this is not universal.
[6.0.2] Theorem: For irreducibles σ, V and τ, W of G and H, the external tensor product σ ⊗ τ is an irreducible of G × H.
Proof: We can put a G × H-invariant hermitian inner product h, i on σ ⊗ τ, by averaging. That is, the action of G × H is by unitary operators.
Then, as in the proof of complete reducibility, the orthogonal complement X⊥ of a G × H-stable subspace X is again G × H-stable. Therefore, both the orthogonal projection to X and the orthogonal projection to X⊥ are G × H maps. They are not scalars. Thus, complementing Schur’s lemma, a reducible unitary representation of G × H would have non-scalar G × H endomorphisms.
To prove σ ⊗τ is irreducible, prove that a G×H-endomorphism T of σ ⊗τ is scalar. For w ∈ W and µ ∈ W ∨ the map ϕw,µ : V → V ⊗ C ≈ V defined by v / v ⊗ w / T (v ⊗ w) / (1 ⊗ µ) T (v ⊗ w) is a G-map. By Schur’s lemma, since σ is irreducible,
ϕw,µ(v) = θw,µ · v (for scalar θw,µ)
∨ ∨ The map W → W by µ → (w → θw,µ) is an H-morphism, since for h ∈ H and 0 6= v ∈ V
−1 θw,hµ · v = (1 ⊗ hµ) T (v ⊗ w) = (1 ⊗ µ)(1V ⊗ h ) T (v ⊗ w)
−1 −1 = (1 ⊗ µ) T (1V ⊗ h )(v ⊗ w) = (1 ⊗ µ) T (v ⊗ h w) = ϕh−1w,µ(v) = θh−1w,µ · v
Since W is irreducible, W ∨ is irreducible, and by Schur’s lemma this map sends µ → cµ for some c ∈ C. That is, there is c such that (w → θw,µ) = c · (w → µ(w))
5 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
∨ ∨ or θw,µ = c · µ(w). For λ ∈ V and µ ∈ W (λ ⊗ µ) T (v ⊗ w) = λ(θw,µ · v) = λ(v) · c · µ(w) = c · hv ⊗ w, λ ⊗ µi
Thus, T acts by the scalar c, and the tensor product is irreducible. ///
[6.0.3] Remark: Ideas from the above argument are re-used in Schur orthogonality and Schur inner-product formulas.
[6.0.4] Theorem: For finite G, H, any irreducible ρ of G × H is isomorphic to σ ⊗ τ for irreducibles σ, τ of G and H.
G×H Proof: By complete reducibility, ρ as a G-representation (properly denoted ResG ρ) contains an irreducible σ of G. The G-homs σ → ρ are the G-invariant C-linear maps from σ → ρ: identifying HomC(σ, ρ) with σ∨ ⊗ ρ, ∨ G HomG(σ, ρ) ≈ (σ ⊗ ρ) (isomorphism of C-vectorspaces) For v ∈ σ, λ ∈ σ∨, and w ∈ ρ, the latter action of G is
(g · (λ ⊗ w))(v) = (gλ ⊗ gw) = (gλ)(v) · gw = λ(g−1v) · gw
The corresponding action on HomG(σ, ρ) is then
(g · ϕ)(v) = g · (ϕ(g−1v))
The C-vectorspace HomG(σ, ρ) is an H-representation, by
(h · ϕ)(v) = h · (ϕ(v)) (for v ∈ σ and h ∈ H)
Let τ be an irreducible subrepresentation of HomG(σ, ρ). From earlier, σ ⊗ τ is an irreducible G × H- representation. The map
T : σ ⊗ τ −→ ρ by T (v ⊗ ϕ) = ϕ(v)
is a G × H-homomorphism. Since ρ is irreducible, and T is not the zero map, it is a surjection. Since σ ⊗ τ is irreducible, T is injective. ///
[6.0.5] Remark: The argument also shows that σ and τ are uniquely determined, up to isomorphism.
7. Matrix coefficient functions
For ρ, V a representation of G, the matrix coefficient function attached to v ∈ V and λ ∈ V ∨ is
ρ ∨ cv,λ(g) = cv,λ(g) = λ(ρ(g)(v) (for g ∈ G, v ∈ V, λ ∈ V )
ρ ∨ 2 [7.0.1] Claim: The map v ⊗ λ → cv,λ is a G × G-map from ρ ⊗ ρ to the biregular representation on L (G).
Proof: The biregular representation’s behavior is
−1 −1 cv,λ(y gx) = λ(y gx)v = (y · λ)(g(x · v) = cxv,yλ(g)
That is, with L the left regular representation of G and R the right regular representation of G on functions on G, we have LyRxcv,λ = cxv,yλ. ///
6 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
8. Schur orthogonality
[8.0.1] Theorem: Matrix coefficient functions attached to non-isomorphic representations are mutually orthogonal in L2(G). That is, for σ, V and τ, W non-isomorphic representations of G, for all v ∈ V , λ ∈ V ∨, w ∈ W , and µ ∈ W ∨, σ τ X σ τ hcv,λ, cw,µiL2(G) = cv,λ(g) cw,µ(g) = 0 g∈G
Proof: Give σ, V a G-invariant inner product. For fixed λ ∈ V ∨, we have a G-map S : V → L2(G) by σ ∨ 2 τ Sv = cv,λ. Similarly, for fixed µ ∈ W we have a G-map T : W → L (G) by T w = cw,µ. Then
σ τ ∗ ∗ 2 hcv,λ, cw,µiL2(G) = hSv, T wiL2(G) = hv, (S ◦ T )wiV (with adjoint-map S : L (G) → V )
The composition S∗ ◦ T maps W → V , and commutes with the action of G. Thus, if S∗ ◦ T were not the zero map it would be a G-isomorphism. But σ 6≈ τ, so S∗ ◦ T = 0, and the inner product is 0. ///
[8.0.2] Remark: A little later, we will give the Schur inner product formula for matrix coefficient functions in the case σ ≈ τ.
o 9. Representations of Cc (G), convolutions
o Functions ϕ ∈ Cc (G) act on representations (ρ, V ) of G by an averaged version of the action:
X X o ϕ · v = ρ(ϕ)(v) = ϕ(g) · gv = ϕ(g) · ρ(g)(v)(v ∈ V , ϕ ∈ Cc (G)) g∈G g∈G
Since finite groups are discrete, the Dirac δ-functions 1 (for g = xo) δxo (g) = 0 (for g 6= xo)
o at xo ∈ G are in Cc (G). The averaged action of such a function is not really averaged: X ρ(δxo )v = δxo (g) · gv = 1 · xov = xov g∈G
o The compactly-supported continuous complex-valued functions Cc (G) on G have a uniquely-determined convolution product ϕ× −→ ϕ ∗ ψ characterized by
ρ(ϕ ∗ ψ) = ρ(ϕ) ◦ ρ(ψ) (for every ρ)
A formula for the convolution follows from this defining property: for fixed v ∈ V , X X (ρ(ϕ) ◦ ρ(ψ))v = ρ(ϕ)(ρ(ψ)v) = ϕ(x) ρ(x) ψ(g) ρ(g)v x∈G g∈G
X X X X = ϕ(x) ψ(g) ρ(xg)v = ϕ(xg−1) ψ(g) ρ(x)v g∈G x∈G x∈G g∈G
7 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
by reversing the order of summation and replacing x by xg−1. This is
X X ϕ(xg−1) ψ(g) ρ(x)v x∈G g∈G
Thus, the inner integral in the latter expression is the convolution (acting on v), that is,
X (ϕ ∗ ψ)(x) = ϕ(xg−1) ψ(g) g∈G
giving the desired ρ(ϕ) ◦ ρ(ψ) = ρ(ϕ ∗ ψ) = ρ(ϕ) ◦ ρ(ψ) (for every ρ)
o Unsurprisingly, convolution is associative, as a consequence of its characterization: for f, ϕ, ψ in Cc (G), X X X (f ∗ (ϕ ∗ ψ))(g) = f(gx−1)(ϕ ∗ ψ)(x) = f(gx−1) ϕ(xy−1) ψ(y) x∈G x∈G y∈G
X X = f(gy−1x−1) ϕ(x) ψ(y) y∈G x∈G by changing the order of summation and replacing x by xy. Then this is
X (f ∗ ϕ)(gy−1) ψ(y) = ((f ∗ ϕ) ∗ ψ)(g) y∈G
as asserted.
o The Dirac δ at 1 ∈ G is the unit in Cc (G) with convolution.
[9.0.1] Proposition: For unitary ρ, the adjoint operator to ρ(ϕ) is ρ(ϕ∗) with
ϕ∗(g) = ϕ(g−1)
Proof: Computing directly,
X X X hρ(ϕ)v, wi = ϕ(g) hρ(g)v, wi = ϕ(g) hv, ρ(g)∗wi = ϕ(g) hv, ρ(g−1)wi g∈G g∈G g∈G by unitariness of ρ. Replacing g by g−1, this is
X X D X E ϕ(g−1) hv, ρ(g)wi = ϕ(g−1) hv, ρ(g)wi = v, ϕ¯(g−1) ρ(g)w = hv, ρ(ϕ∗)wi g∈G g∈G g∈G as claimed, where the complex conjugate appears because the inner product is conjugate-linear in its second argument. ///
8 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
o 10. G-representations versus Cc (G)-modules
o The category of group representations of G and G-maps among them is the same as the category of Cc (G)- o modules and Cc (G)-modules homomorphisms: for representations (ρ, V ) and (τ, W ) of G,
o • The collection of G-stable subspaces of V is identical to the collection of Cc (G)-stable subspaces of V .
o • A linear map T : V → W is a G-homomorphism if and only if it is an Cc (G)-module homomorphism.
o • The representation (ρ, V ) is G-irreducible if and only if it is Cc (G)-irreducible. That is, these two categories are equivalent, in the sense that the objects are in bijection, and the morphisms are in natural bijection.
11. Decomposition of biregular representation on L2(G)
For irreducible ρ of G, formation of matrix coefficient functions
∨ 2 ρ ⊗ ρ → L (G) by v ⊗ λ → cv,λ
is a G × G map. The Schur orthogonality relations demonstrate the mutual orthogonality of the images for non-isomorphic irreducibles ρ. For ρ an irreducible unitary representation of G, let L2(G)ρ denote the ρ-isotypic component inside L2(G) under the right regular representation R, that is, the sum of all copies of ρ in L2(G) with right regular representation of G. Certainly L2(G)ρ ⊃ ρ ⊗ ρ∨.
[11.0.1] Theorem: The isotypic component L2(G)ρ under the right regular representation is stable under the left regular representation L, and as G × G-representation with the biregular representation,
L2(G)ρ ≈ ρ ⊗ ρ∨
Therefore, as a G × G representation,
M L2(G) ≈ ρ ⊗ ρ∨ (sum over G-irreducibles ρ) ρ
Proof: Again Schur orthogonality gives directness of the sum, in
M L2(G) ⊃ ρ ⊗ ρ∨ (sum over G-irreducibles ρ) ρ
It remains to be shown that there is nothing else in L2(G). By complete reducibility, it suffices to show that the ρ-isotype under the right regular representation is no larger than the image of ρ ⊗ ρ∨.
2 ρ ∨ For f in a copy of ρ inside L (G) orthogonal to all coefficient functions cv,λ coming from ρ ⊗ ρ , by the stability of matrix coefficient functions under the left regular representation and the unitariness of the biregular representation, all left translates Lgf are still orthogonal to all these coefficient functions.
9 Paul Garrett: Generalities on representations of finite groups (October 6, 2014)
o ∨ −1 With real-valued ϕ ∈ Cc (G), let ϕ (g) = ϕ(g ). By the equivalence of categories of G-representations o and Cc (G)-modules, there is ϕ such that X X 0 6= ϕ∨ · f = g −→ f(h−1g) ϕ∨(h) = g −→ f(hg) ϕ(h) h∈G h∈G
With η the orthogonal projection of ϕ to the ρ-isotype,
∨ ϕ f = cf,ϕ = cf,η and ∨ ρ ρ ρ hρ(ϕ )f, cv,λi = hcf,η, cv,λi = In particular, taking v = f and λ = η gives a contradiction: there is no non-trivial f in the ρ-isotype orthogonal to all the matrix coefficient functions. That is, ρ ⊗ ρ∨ is the whole ρ-isotype. ///
10