(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 ' :(σ; V ) −! (τ; W ) from one G-representation (σ; V ) to another (τ; W ) is, as expected, a vector-space map ' : V ! W which commutes with or respects the action of G in the natural sense: ' ◦ σ(g) = τ(g) ◦ ' (for all g 2 G) Vector-wise, using the lighter notation, the requirement is '(g · v) = g · '(v) (for all v 2 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 2 G and W 2 W , ρ(g)(W ) 2 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 f0g 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 2 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 2 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 g2G By design, this is G-invariant: for h 2 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 g2G g2G g2G 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 2 C, 1 X 1 X hav + v0; wi = hg(av + v0); gwi = ahgv; gwi + hgv0; gwi #G o #G o o g2G g2G 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 g2G g2G 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 2 W 0 and w 2 W ) since g−1W 2 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 2 V _, a 2 C, and v 2 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 2 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 2 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 2 G and f 2 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 2 G and f 2 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 2 L2(G), and g 2 G, using the definition and changing variables, X X hRgf; RgF i = f(xg) F (xg) = f(x) F (x) = hf; F i x2G x2G proving the unitariness.