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Representation Theory 1 LINEAR AND UNITARY GROUPS Representation theory 1 Linear and unitary groups Let V denote a linear space with eld of scalars F (usually either the real or the complex numbers). The general linear group GL(V )=fA: V !V j det A 6= 0g 1 LINEAR AND UNITARY GROUPS over V consist of all invertible linear operators on V , with product the composition (multiplication) of operators. If V has nite dimension n, then for each choice of basis B=fe1;:::; eng there exist (non-canonical) isomorphisms ΓB : GL(V )!GLn(F) n X A(ei) = ΓB(A)ij ej j=1 If 0 0 0 is another basis of , then B =fe1;:::; eng V −1 ΓB0 (A) = C ΓB(A) C where 0 P (basis change). ei = j Cijej matrices 6= linear operators 1 LINEAR AND UNITARY GROUPS Subgroups of GL(V ) are termed linear groups. 1 LINEAR AND UNITARY GROUPS Examples: 1. the group × Sc(V ) = fλidV j λ2F g of scalar operators, isomorphic to the unit group F×; 2. the special linear group SL(V ) = fA2GL(V ) j det A=1g of unimodular (i.e. having determinant 1) operators; 3. the stabilizer subgroup Stab(v) = fA2GL(V ) j Av =vg of a vector v 2V . 1 LINEAR AND UNITARY GROUPS Linear groups are more amenable to study because · linear algebra methods (e.g. spectral decomposition, determinants, etc.) are available in the characterization of their elements; · there exist group constructions specic to linear groups (linear du- ality, tensor products, etc.) · special algorithms for collections of linear operators. Linear groups are rather the exception than the rule, but some special classes of groups (like nite, compact Lie, etc.) may be shown to have all members isomorphic to some linear group. 1 LINEAR AND UNITARY GROUPS A subset W ⊆ V of a linear space V (with eld of scalars F) is a linear subspace if x+y 2W and αx2W whenever x; y 2W and α2F. A linear subspace W ⊆V is nontrivial if it is dierent from the zero subspace and the whole space, i.e. 0<dim W <dim V . The translates x+W =fx+y j y 2W g of a linear subspace W ⊆V (where x runs over the elements of V ) form a linear space, the factor space V=W , with addition and multiplication by a scalar dened as (x+W ) + (y+W ) = (x+y)+W α(x+W ) = αx+W for x; y 2V and α2F. 1 LINEAR AND UNITARY GROUPS Remark. Any basis BW of a linear subspace W ⊆ V can be completed (in many dierent ways) to a basis BV ⊇ BW of the whole space, and each element of the dierence set BV nBW corresponds to a basis vector of the factor space. Consequence: dim (V=W )=dim V − dim W A linear subspace W ⊆ V is an invariant subspace of the linear group G < GL(V ) if all group elements map it onto itself, i.e. gx 2 W for all g 2G and x2W . Remark. The zero subspace and the whole space are always invariant. 1 LINEAR AND UNITARY GROUPS Given an invariant subspace W ⊆V of G<GL(V ), the restrictions gW : W ! W x 7! gx to W of the group elements and the factored operators g=W : V=W ! V=W x + W 7! gx + W are well dened linear operators making up linear groups, the reduction GW = fgW j g 2Gg < GL(W ) and the factor G=W = fg=W j g 2Gg < GL(V=W ) of the linear group G. 1 LINEAR AND UNITARY GROUPS Remark. Given a basis BW of W and its completion BV to one of V , the representation matrices with respect to the latter read ! ΓBW (g) T (g) ΓBV (g) = 0 ΓBVnBW (g) for some matrices T (g) that satisfy T (gh) = ΓBW (g) T (h) + T (g) ΓBVnBW (h) for all g; h2G, i.e. they are upper triangular block matrices. A linear group is reducible if it has a nontrivial invariant subspace, oth- erwise it is called irreducible. Schur's lemma: any operator that commutes with all elements of an irreducible linear group is a multiple of the identity operator. 1 LINEAR AND UNITARY GROUPS A unitary operator U on a Hilbert-space H (a complex linear space endowed with a positive denite inner product, complete with respect to the norm topology) is a linear operator preserving the inner product, i.e. hUx; Uyi = hx; yi for all x; y 2H. Remark. An antiunitary operator A is one which is antilinear, i.e. A (αx+βy) = αAx + βAy for all α; β 2C and x; y 2H, and satises hAx; Ayi = hx; yi (with the bar denoting complex conjugation). 1 LINEAR AND UNITARY GROUPS The product of unitary operators is also unitary unitary operators form a group U(H), the unitary group of H. Remark. If H is of nite dimension n=dim H, then one usually denotes the unitary group U(H) by U(n). Any unitary operator can be diagonalized, and its eigenvalues lie on the complex unit circle U(1)=fz 2C j zz =1g. The product of two antiunitary operators is unitary, and any two of them dier by a unitary operator antiunitary operators form a coset of U(H), hence it is enough to understand the structure of the latter. 1 LINEAR AND UNITARY GROUPS Unitary groups in physics: quantum systems and gauge symmetries. observable quantities ! self-adjoint operators on Hilbert-space energy ! Hamiltonian (generator of time translations) Wigner's theorem: the symmetries of a quantum system correspond to (anti-)unitary operators commuting with its Hamiltonian. Remark. Antiunitary operators from time reversal symmetry (exchang- ing cause and eect). Gauge symmetries of fundamental interactions described by (special) unitary groups in the Standard Model. 2 LINEAR REPRESENTATIONS 2 Linear representations Problem: given an abstract group, nd a linear group isomorphic to it. No solution for a generic group need to look for suitable (linear) homomorphic images. A (linear) representation of the group G over the linear space V is a homomorphism D :G!GL(V ) (the representation is called faithful when its kernel is trivial). The image of a representation is always a linear group, which is isomor- phic to G precisely if the representation is faithful. 2 LINEAR REPRESENTATIONS The eld of scalars and the dimension of D equal those of the represen- tation space V , in particular dim D = dim V The theory becomes pretty complicated for generic eld of scalars best to look at representations over the complex numbers C. Remark. For each choice of a basis B=fe1;:::; eng of V , the linear rep- resentation D : G ! GL(V ) determines a degree n matrix representation via the rule , where DB : G!GLn(F) DB =ΓB ◦D ΓB : GL(V )!GLn(F) is the isomorphism associated to B. 2 LINEAR REPRESENTATIONS Examples: 1. For any group G and linear space V , the map 1V : G ! GL(V ) g 7! idV is a representation, the trivial representation of G over V ; 2. For a linear group G<GL(V ), the inclusion map DG : G ! GL(V ) g 7! g is a representation of G, the dening representation; 2 LINEAR REPRESENTATIONS 3. Given a eld F, the ring F(X)=ff :X !Fg of F-valued functions on the set X (with pointwise addition and multiplication) is a linear space over F of dimension X . Then the assignment D(π): F(X) ! F(X) f 7! f ◦ π−1 denes a linear operator acting on F(X), and for each subgroup G<Sym(X), the map D : G ! GL(F(X)) π 7! D(π) is a representation of G on F(X), the permutation representation associated to the action of G on X. 2 LINEAR REPRESENTATIONS 4. Given an action α: G!Aut(X; µ) of G on the measure space (X; µ) (i.e. an action of G on X for which µ is an invariant measure, that is µ(gS)=µ(S) for every g 2G and every measurable set S ⊆X), consider the Hilbert-space L2(X; µ) of square integrable functions on X. Because the measure µ is invariant, the maps L(g): L2(X; µ) ! L2(X; µ) 7! ◦ αg−1 are unitary operators that satisfy L(gh) = L(g)◦L(h) for g; h 2 G, hence they dene a (unitary) representation of G on L2(X; µ), the (left) regular representation. 2 LINEAR REPRESENTATIONS A unitary representation of G on a Hilbert-space H is a homomorphism U : G ! U(H) into the group of unitary operators on H, i.e. a linear representation all of whose representation operators are unitary: hU(g)x; U(g)yi = hx; yi for all x; y 2H and g 2G. A unitarizable representation D : G!GL(V ) is one for which there exists a positive denite scalar product on V (making it a Hilbert-space) for which all representation operators are unitary. All representations of a nite or compact group are unitarizable. 3 EQUIVALENCE AND REDUCIBILITY 3 Equivalence and reducibility The representations D1 : G!GL(V1) and D2 : G!GL(V2) are (linearly) ∼ equivalent, denoted D1 = D2, if there exists an invertible linear map A: V1 !V2 such that D2(g)A=AD1(g) for all g 2G. Remark. The dimension of equivalent representations is the same. Linear equivalence is a reexive, symmetric and transitive relation. Equivalent representations are practically the same, e.g. the representa- tion matrices of group elements coincide (w.r.t. suitable bases). Problem: classify all representations of a given group (up to equivalence).
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