Schur orthogonality relations

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).

Finite groups

edit

Intrinsic statement

edit

The space of complex-valued class functions of a finite group G has a natural inner product:

 

where   denotes the complex conjugate of the value of   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:

 

For  , applying the same inner product to the columns of the character table yields:

 

where the sum is over all of the irreducible characters   of  , and   denotes the order of the centralizer of  . 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

edit

Let   be a matrix element of an irreducible matrix representation   of a finite group   of order |G|. Since it can be proven that any matrix representation of any finite group is equivalent to a unitary representation, we assume   is unitary:

 

where   is the (finite) dimension of the irreducible representation  .[1]

The orthogonality relations, only valid for matrix elements of irreducible representations, are:

 

Here   is the complex conjugate of   and the sum is over all elements of G. The Kronecker delta   is 1 if the matrices are in the same irreducible representation  . If   and   are non-equivalent it is zero. The other two Kronecker delta's state that the row and column indices must be equal (  and  ) 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 to 1). This is an irreducible representation. The great orthogonality relations immediately imply that

 

for   and any irreducible representation   not equal to the identity representation.

Example of the permutation group on 3 objects

edit

The 3! permutations of three objects form a group of order 6, commonly denoted S3 (the symmetric group of degree three). This group is isomorphic to the point group  , 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   and in the case of   one usually writes  . In both cases the representation consists of the following six real matrices, each representing a single group element:[2]

 

The normalization of the (1,1) element:

 

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:

 

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

edit

The trace of a matrix is a sum of diagonal matrix elements,

 

The collection of traces is the character   of a representation. Often one writes for the trace of a matrix in an irreducible representation with character  

 

In this notation we can write several character formulas:

 

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

 

which helps us to determine how often the irreducible representation   is contained within the reducible representation   with character  .

For instance, if

 

and the order of the group is

 

then the number of times that   is contained within the given reducible representation   is

 

See Character theory for more about group characters.

Compact groups

edit

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   has unique bi-invariant Haar measure, so that the volume of the group is 1. Denote this measure by  . Let   be a complete set of irreducible representations of  , and let   be a matrix coefficient of the representation  . The orthogonality relations can then be stated in two parts:

1) If   then

 

2) If   is an orthonormal basis of the representation space   then

 

where   is the dimension of  . 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)

edit

An example of an r = 3 parameter group is the matrix group SO(3) consisting of all 3 × 3 orthogonal matrices with unit determinant. A possible parametrization of this group is in terms of Euler angles:   (see e.g., this article for the explicit form of an element of SO(3) in terms of Euler angles). The bounds are   and  .

Not only the recipe for the computation of the volume element   depends on the chosen parameters, but also the final result, i.e. the analytic form of the weight function (measure)  .

For instance, the Euler angle parametrization of SO(3) gives the weight   while the n, ψ parametrization gives the weight   with  

It can be shown that the irreducible matrix representations of compact Lie groups are finite-dimensional and can be chosen to be unitary:

 

With the shorthand notation

 

the orthogonality relations take the form

 

with the volume of the group:

 

As an example we note that the irreducible representations of SO(3) are Wigner D-matrices  , which are of dimension  . Since

 

they satisfy

 

Notes

edit
  1. ^ The finiteness of   follows from the fact that any irreducible representation of a finite group G is contained in the regular representation.
  2. ^ This choice is not unique; any orthogonal similarity transformation applied to the matrices gives a valid irreducible representation.

References

edit

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.