- Frobenius-Schur indicator
mathematicsthe Schur indicator, named after Issai Schur, or Frobenius-Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has, and can be used to classify the irreducible representations of compact groups on real vector spaces.
If a representation of a compact group "G" has character χ its Frobenius-Schur indicator is defined to be
Haar measureμ with μ("G") = 1. When "G" is finite it is given by
It provides criterion (for
compact groups "G") for reality of irreducible representations in terms of character theory. This will be discussed below in the case of finite groups, but the general compact case is completely analogous.
Real irreducible representations
There are 3 types of irreducible real representations of a finite group on a real vector space "V", as the ring of endomorphisms commuting with the group action can be isomorphic to either the real numbers, or the complex numbers, or the quaternions.
*If the ring is the real numbers, then "V"⊗C is an irreducible complex representation with Schur indicator 1, also called a
*If the ring is the complex numbers, then "V" has two different conjugate complex structures, giving two irreducible complex representations with Schur indicator 0, sometimes called
*If the ring is the quaternions numbers, then choosing a subring of the quaternions isomorphic to the complex numbers makes "V" into an irreducible complex representation of "G" with Schur indicator − 1, called a
Moreover every irreducible representation on a complex vector space can be constructed from a unique irreducible representation on a real vector space in one of the three ways above. So knowing the irreducible representations on complex spaces and their Schur indicators allows one to read off the irreducible representations on real spaces.
Real representations can be complexified to get a complex representation of the same dimension and complex representations can be converted into a real representation of twice the dimension by treating the real and imaginary components separately. Also, since all finite dimensional complex representations can be turned into a unitary representation for unitary representations, the dual representation is also a (complex) conjugate representation because the Hilbert space norm gives an
antilinear bijectivemap from the representation to its dual representation.
Self-dual complex irreducible representation correspond to either real irreducible representation of the same dimension or real irreducible representations of twice the dimension called
quaternionic representations (but not both) and non-self-dual complex irreducible representation correspond to a real irreducible representation of twice the dimension. Note for the latter case, both the complex irreducible representation and its dual give rise to the same real irreducible representation. An example of a quaternionic representation would be the four dimensional real irreducible representation of the quaternion group"Q"8.
Invariant bilinear forms
If "V" is the underlying vector space of a representation, then
can be decomposed as the direct sum of two subrepresentations, the "symmetric tensor product"
and the "antisymmetric tensor product"
It's easy to show that
using a basis set.
is a self-intertwiner, for any integer "n",
is also a self-intertwiner. By Schur's lemma, this will be a multiple of the identity for irreducible representations. The trace of this self-intertwiner is called the nth "Frobenius-Schur indicator".
The original case of the Frobenius-Schur indicator is that for "n" = 2. The zeroth indicator is the dimension of the irreducible representation, the first indicator would be 1 for the trivial representation and zero for the other irreducible representations.
It resembles the
Casimir invariants for Lie algebrairreducible representations. In fact, since any rep of G can be thought of as a module for C ["G"] and vice versa, we can look at the center of C ["G"] . This is analogous to looking at the center of the universal enveloping algebraof a Lie algebra. It is simple to check that
belongs to the center of C ["G"] , which is simply the subspace of class functions on "G".
*cite book | author=Jean-Pierre, Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | id=ISBN 0-387-90190-6
Wikimedia Foundation. 2010.
Look at other dictionaries:
Ferdinand Georg Frobenius — Infobox Scientist name = PAGENAME box width = image size =150px caption = PAGENAME birth date = October 26, 1849 birth place = Charlottenburg death date = August 3, 1917 death place = Berlin residence = citizenship = nationality = German… … Wikipedia
Issai Schur — (January 10, 1875 in Mogilyov ndash; January 10, 1941 in Tel Aviv) was a mathematician who worked in Germany for most of his life. He studied at Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at Bonn,… … Wikipedia
Real representation — In the mathematical field of representation theory a real representation is usually a representation on a real vector space U , but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an… … Wikipedia
List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… … Wikipedia
Quaternionic representation — In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map:jcolon V o V, which satisfies:j^2=… … Wikipedia
Complex representation — The term complex representation has slightly different meanings in mathematics and physics. In mathematics, a complex representation is a group representation of a group (or Lie algebra) on a complex vector space. In physics, a complex… … Wikipedia
Symplectic representation — In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space ( V , omega; ) which preserves the symplectic form omega; . Here omega; is a nondegenerate… … Wikipedia