In mathematics, the Schur orthogonality relations, which were proven by
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the ...
through
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations
of a group ' ...
, express a central fact about
representations
''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of finite
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
.
They admit a generalization to the case of
compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s in general, and in particular
compact Lie groups, such as the
rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
.
Finite groups
Intrinsic statement
The space of complex-valued
class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjugat ...
s of a finite group G has a natural
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
:
:
where
means the complex conjugate of the value of
on ''g''. With respect to this inner product, the irreducible
characters
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
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 ''G'' and the symbol
denotes the order of the
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of
. Note that since and 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
be a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
element of an
irreducible
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
matrix representation
Matrix representation is a method used by a computer language to store matrix (mathematics), matrices of more than one dimension in computer storage, memory.
Fortran and C (programming language), C use different schemes for their native arrays. Fo ...
of a finite group
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 In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' ...
, we assume
is unitary:
:
where
is the (finite) dimension of the irreducible representation
.
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
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
is unity 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 onto the real number 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
The 3! permutations of three objects form a group of order 6, commonly denoted (
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
). This group is isomorphic to the
point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
, 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 one usually labels this representation
by the
Young tableau In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups a ...