Schur orthogonality relations

Schur orthogonality relations   (Redirected from Great orthogonality theorem) Jump to navigation Jump to search In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

Contents 1 Finite groups 1.1 Intrinsic statement 1.2 Coordinates statement 1.3 Example of the permutation group on 3 objects 1.4 Direct implications 2 Compact Groups 2.1 An Example SO(3) 3 Notes 4 References Finite groups Intrinsic statement The space of complex-valued class functions of a finite group G has a natural inner product: {displaystyle leftlangle alpha ,beta rightrangle :={frac {1}{left|Gright|}}sum _{gin G}alpha (g){overline {beta (g)}}} where {displaystyle {overline {beta (g)}}} means the complex conjugate of the value of {displaystyle beta } on g. With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table: {displaystyle leftlangle chi _{i},chi _{j}rightrangle ={begin{cases}0&{mbox{ if }}ineq j,\1&{mbox{ if }}i=j.end{cases}}} For {displaystyle g,hin G} , applying the same inner product to the columns of the character table yields: {displaystyle sum _{chi _{i}}chi _{i}(g){overline {chi _{i}(h)}}={begin{cases}left|C_{G}(g)right|,&{mbox{ if }}g,h{mbox{ are conjugate }}\0&{mbox{ otherwise.}}end{cases}}} where the sum is over all of the irreducible characters {displaystyle chi _{i}} of G and the symbol {displaystyle left|C_{G}(g)right|} denotes the order of the centralizer of {displaystyle g} . Note that since g and h are conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.

The orthogonality relations can aid many computations including: decomposing an unknown character as a linear combination of irreducible characters; constructing the complete character table when only some of the irreducible characters are known; finding the orders of the centralizers of representatives of the conjugacy classes of a group; and finding the order of the group. Coordinates statement Let {displaystyle Gamma ^{(lambda )}(R)_{mn}} be a matrix element of an irreducible matrix representation {displaystyle Gamma ^{(lambda )}} of a finite group {displaystyle G={R}} of order |G|, i.e. G has |G| elements. Since it can be proven that any matrix representation of any finite group is equivalent to a unitary representation, we assume {displaystyle Gamma ^{(lambda )}} is unitary: {displaystyle sum _{n=1}^{l_{lambda }};Gamma ^{(lambda )}(R)_{nm}^{*};Gamma ^{(lambda )}(R)_{nk}=delta _{mk}quad {hbox{for all}}quad Rin G,} where {displaystyle l_{lambda }} is the (finite) dimension of the irreducible representation {displaystyle Gamma ^{(lambda )}} .[1] The orthogonality relations, only valid for matrix elements of irreducible representations, are: {displaystyle sum _{Rin G}^{|G|};Gamma ^{(lambda )}(R)_{nm}^{*};Gamma ^{(mu )}(R)_{n'm'}=delta _{lambda mu }delta _{nn'}delta _{mm'}{frac {|G|}{l_{lambda }}}.} Here {displaystyle Gamma ^{(lambda )}(R)_{nm}^{*}} is the complex conjugate of {displaystyle Gamma ^{(lambda )}(R)_{nm},} and the sum is over all elements of G. The Kronecker delta {displaystyle delta _{lambda mu }} is unity if the matrices are in the same irreducible representation {displaystyle Gamma ^{(lambda )}=Gamma ^{(mu )}} . If {displaystyle Gamma ^{(lambda )}} and {displaystyle Gamma ^{(mu )}} are non-equivalent it is zero. The other two Kronecker delta's state that the row and column indices must be equal ( {displaystyle n=n'} and {displaystyle m=m'} ) in order to obtain a non-vanishing result. This theorem is also known as the Great (or Grand) Orthogonality Theorem.

Every group has an identity representation (all group elements mapped onto the real number 1). This is an irreducible representation. The great orthogonality relations immediately imply that {displaystyle sum _{Rin G}^{|G|};Gamma ^{(mu )}(R)_{nm}=0} for {displaystyle n,m=1,ldots ,l_{mu }} and any irreducible representation {displaystyle Gamma ^{(mu )},} not equal to the identity representation.

Example of the permutation group on 3 objects The 3! permutations of three objects form a group of order 6, commonly denoted S3 (symmetric group). This group is isomorphic to the point group {displaystyle C_{3v}} , consisting of a threefold rotation axis and three vertical mirror planes. The groups have a 2-dimensional irreducible representation (l = 2). In the case of S3 one usually labels this representation by the Young tableau {displaystyle lambda =[2,1]} and in the case of {displaystyle C_{3v}} one usually writes {displaystyle lambda =E} . In both cases the representation consists of the following six real matrices, each representing a single group element:[2] {displaystyle {begin{pmatrix}1&0\0&1\end{pmatrix}}quad {begin{pmatrix}1&0\0&-1\end{pmatrix}}quad {begin{pmatrix}-{frac {1}{2}}&{frac {sqrt {3}}{2}}\{frac {sqrt {3}}{2}}&{frac {1}{2}}\end{pmatrix}}quad {begin{pmatrix}-{frac {1}{2}}&-{frac {sqrt {3}}{2}}\-{frac {sqrt {3}}{2}}&{frac {1}{2}}\end{pmatrix}}quad {begin{pmatrix}-{frac {1}{2}}&{frac {sqrt {3}}{2}}\-{frac {sqrt {3}}{2}}&-{frac {1}{2}}\end{pmatrix}}quad {begin{pmatrix}-{frac {1}{2}}&-{frac {sqrt {3}}{2}}\{frac {sqrt {3}}{2}}&-{frac {1}{2}}\end{pmatrix}}} The normalization of the (1,1) element: {displaystyle sum _{Rin G}^{6};Gamma (R)_{11}^{*};Gamma (R)_{11}=1^{2}+1^{2}+left(-{tfrac {1}{2}}right)^{2}+left(-{tfrac {1}{2}}right)^{2}+left(-{tfrac {1}{2}}right)^{2}+left(-{tfrac {1}{2}}right)^{2}=3.} In the same manner one can show the normalization of the other matrix elements: (2,2), (1,2), and (2,1). The orthogonality of the (1,1) and (2,2) elements: {displaystyle sum _{Rin G}^{6};Gamma (R)_{11}^{*};Gamma (R)_{22}=1^{2}+(1)(-1)+left(-{tfrac {1}{2}}right)left({tfrac {1}{2}}right)+left(-{tfrac {1}{2}}right)left({tfrac {1}{2}}right)+left(-{tfrac {1}{2}}right)^{2}+left(-{tfrac {1}{2}}right)^{2}=0.} Similar relations hold for the orthogonality of the elements (1,1) and (1,2), etc. One verifies easily in the example that all sums of corresponding matrix elements vanish because of the orthogonality of the given irreducible representation to the identity representation.

Direct implications The trace of a matrix is a sum of diagonal matrix elements, {displaystyle operatorname {Tr} {big (}Gamma (R){big )}=sum _{m=1}^{l}Gamma (R)_{mm}.} The collection of traces is the character {displaystyle chi equiv {operatorname {Tr} {big (}Gamma (R){big )};|;Rin G}} of a representation. Often one writes for the trace of a matrix in an irreducible representation with character {displaystyle chi ^{(lambda )}} {displaystyle chi ^{(lambda )}(R)equiv operatorname {Tr} left(Gamma ^{(lambda )}(R)right).} In this notation we can write several character formulas: {displaystyle sum _{Rin G}^{|G|}chi ^{(lambda )}(R)^{*},chi ^{(mu )}(R)=delta _{lambda mu }|G|,} which allows us to check whether or not a representation is irreducible. (The formula means that the lines in any character table have to be orthogonal vectors.) And {displaystyle sum _{Rin G}^{|G|}chi ^{(lambda )}(R)^{*},chi (R)=n^{(lambda )}|G|,} which helps us to determine how often the irreducible representation {displaystyle Gamma ^{(lambda )}} is contained within the reducible representation {displaystyle Gamma ,} with character {displaystyle chi (R)} .

For instance, if {displaystyle n^{(lambda )},|G|=96} and the order of the group is {displaystyle |G|=24,} then the number of times that {displaystyle Gamma ^{(lambda )},} is contained within the given reducible representation {displaystyle Gamma ,} is {displaystyle n^{(lambda )}=4,.} See Character theory for more about group characters.

Compact Groups The generalization of the orthogonality relations from finite groups to compact groups (which include compact Lie groups such as SO(3)) is basically simple: Replace the summation over the group by an integration over the group.

Every compact group {displaystyle G} has unique bi-invariant Haar measure, so that the volume of the group is 1. Denote this measure by {displaystyle dg} . Let {displaystyle (pi ^{alpha })} be a complete set of irreducible representations of {displaystyle G} , and let {displaystyle phi _{v,w}^{alpha }(g)=langle v,pi ^{alpha }(g)wrangle } be a matrix coefficient of the representation {displaystyle pi ^{alpha }} . The orthogonality relations can then be stated in two parts: 1) If {displaystyle pi ^{alpha }ncong pi ^{beta }} then {displaystyle int _{G}phi _{v,w}^{alpha }(g)phi _{v',w'}^{beta }(g)dg=0} 2) If {displaystyle {e_{i}}} is an orthonormal basis of the representation space {displaystyle pi ^{alpha }} then {displaystyle int _{G}phi _{e_{i},e_{j}}^{alpha }(g){overline {phi _{e_{m},e_{n}}^{alpha }(g)}}dg=delta _{i,m}delta _{j,n}{frac {1}{d^{alpha }}}} where {displaystyle d^{alpha }} is the dimension of {displaystyle pi ^{alpha }} . These orthogonality relations and the fact that all of the representations have finite dimensions are consequences of the Peter–Weyl theorem.

An Example SO(3) An example of an r = 3 parameter group is the matrix group SO(3) consisting of all 3 x 3 orthogonal matrices with unit determinant. A possible parametrization of this group is in terms of Euler angles: {displaystyle mathbf {x} =(alpha ,beta ,gamma )} (see e.g., this article for the explicit form of an element of SO(3) in terms of Euler angles). The bounds are {displaystyle 0leq alpha ,gamma leq 2pi } and {displaystyle 0leq beta leq pi } .

Not only the recipe for the computation of the volume element {displaystyle omega (mathbf {x} ),dx_{1}dx_{2}cdots dx_{r}} depends on the chosen parameters, but also the final result, i.e. the analytic form of the weight function (measure) {displaystyle omega (mathbf {x} )} .

For instance, the Euler angle parametrization of SO(3) gives the weight {displaystyle omega (alpha ,beta ,gamma )=sin !beta ,,} while the n, ψ parametrization gives the weight {displaystyle omega (psi ,theta ,phi )=2(1-cos psi )sin !theta ,} with {displaystyle 0leq psi leq pi ,;;0leq phi leq 2pi ,;;0leq theta leq pi .} It can be shown that the irreducible matrix representations of compact Lie groups are finite-dimensional and can be chosen to be unitary: {displaystyle Gamma ^{(lambda )}(R^{-1})=Gamma ^{(lambda )}(R)^{-1}=Gamma ^{(lambda )}(R)^{dagger }quad {hbox{with}}quad Gamma ^{(lambda )}(R)_{mn}^{dagger }equiv Gamma ^{(lambda )}(R)_{nm}^{*}.} With the shorthand notation {displaystyle Gamma ^{(lambda )}(mathbf {x} )=Gamma ^{(lambda )}{Big (}R(mathbf {x} ){Big )}} the orthogonality relations take the form {displaystyle int _{x_{1}^{0}}^{x_{1}^{1}}cdots int _{x_{r}^{0}}^{x_{r}^{1}};Gamma ^{(lambda )}(mathbf {x} )_{nm}^{*}Gamma ^{(mu )}(mathbf {x} )_{n'm'};omega (mathbf {x} )dx_{1}cdots dx_{r};=delta _{lambda mu }delta _{nn'}delta _{mm'}{frac {|G|}{l_{lambda }}},} with the volume of the group: {displaystyle |G|=int _{x_{1}^{0}}^{x_{1}^{1}}cdots int _{x_{r}^{0}}^{x_{r}^{1}}omega (mathbf {x} )dx_{1}cdots dx_{r}.} As an example we note that the irreducible representations of SO(3) are Wigner D-matrices {displaystyle D^{ell }(alpha beta gamma )} , which are of dimension {displaystyle 2ell +1} . Since {displaystyle |mathrm {SO} (3)|=int _{0}^{2pi }dalpha int _{0}^{pi }sin !beta ,dbeta int _{0}^{2pi }dgamma =8pi ^{2},} they satisfy {displaystyle int _{0}^{2pi }int _{0}^{pi }int _{0}^{2pi }D^{ell }(alpha beta gamma )_{nm}^{*};D^{ell '}(alpha beta gamma )_{n'm'};sin !beta ,dalpha ,dbeta ,dgamma =delta _{ell ell '}delta _{nn'}delta _{mm'}{frac {8pi ^{2}}{2ell +1}}.} Notes ^ The finiteness of {displaystyle l_{lambda }} follows from the fact that any irreducible representation of a finite group G is contained in the regular representation. ^ This choice is not unique, any orthogonal similarity transformation applied to the matrices gives a valid irreducible representation. References Any physically or chemically oriented book on group theory mentions the orthogonality relations. The following more advanced books give the proofs: M. Hamermesh, Group Theory and its Applications to Physical Problems, Addison-Wesley, Reading (1962). (Reprinted by Dover). W. Miller, Jr., Symmetry Groups and their Applications, Academic Press, New York (1972). J. F. Cornwell, Group Theory in Physics, (Three volumes), Volume 1, Academic Press, New York (1997).

The following books give more mathematically inclined treatments: Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. New York: Springer-Verlag. pp. 13-20. ISBN 0387901906. ISSN 0072-5285. OCLC 2202385. Sengupta, Ambar N. (2012). Representing Finite Groups, A Semisimple Introduction. Springer. ISBN 978-1-4614-1232-8. OCLC 875741967. Categories: Representation theory of groups