Editing Frobenius–Schur indicator

In [[mathematics]], and especially the discipline of [[representation theory]], the '''Schur indicator''', named after [[Issai Schur]], or '''Frobenius–Schur indicator''' describes what invariant bilinear forms a given [[irreducible representation|irreducible]] [[representation theory|representation]] of a [[compact group|compact]] [[group (mathematics)|group]] on a [[complex number|complex]] [[vector space]] has. It can be used to classify the  irreducible representations of compact groups on [[real number|real]] vector spaces.

==Definition==

If a finite-dimensional continuous representation of a compact group ''G'' has [[character theory|character]] &chi; its '''Frobenius–Schur indicator''' is defined to be

:<math>\int_{g\in G}\chi(g^2)\,d\mu</math>

for [[Haar measure]] μ with μ(''G'') = 1. When ''G'' is [[finite group|finite]] it is given by

:<math>{1\over |G|}\sum_{g\in G}\chi(g^2).</math>

If &chi; is a complex irreducible representation, then its Frobenius–Schur indicator is 1, 0, or −1.  It provides a criterion for deciding whether a real irreducible representation of ''G'' is real, complex or [[quaternion]]ic, in a specific sense defined below. Much of the content below discusses the case of finite groups, but the general compact case is analogous.

==Real irreducible representations==
{{Main|Real representation}}

There are ''three types'' of irreducible real representations  of a finite group on a real vector space ''V'', as [[Schur's lemma]] implies that the [[endomorphism ring]] commuting with the group action is a real associative [[division algebra]] and by the [[Frobenius theorem (real division algebras)|Frobenius theorem]] can only be isomorphic to either the real numbers, or the complex numbers, or the quaternions.

*If the ring is the real numbers, then ''V''&otimes;'''C''' is an irreducible complex representation with Schur indicator 1, also called a real representation.
*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 [[complex representation]]s.
*If the ring is the [[quaternion]]s, then choosing a subring of the quaternions isomorphic to the complex numbers makes ''V'' into an irreducible complex representation of ''G'' with Schur indicator &minus;1, called a [[quaternionic representation]].

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]] [[bijective]] map 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 representation]]s (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''<sub>8</sub>.

==Definition in terms of the symmetric and alternating square==
{{See also|Symmetric and alternating squares}}

If {{var|V}} is the underlying vector space of a representation of a group {{var|G}}, then the [[Tensor product of representations|tensor product representation]] <math>V\otimes V</math> can be decomposed as the direct sum of two [[subrepresentation]]s, the '''symmetric square''', denoted <math>\operatorname{Sym}^2(V)</math> (also often denoted by <math>V\otimes_S V</math> or <math>V\odot V</math>) and the '''alternating square''', <math>\operatorname{Alt}^2(V) </math>(also often denoted by <math>\wedge^2V</math>, <math>V\otimes_A V</math>, or <math>V \wedge V</math>).{{sfn|Serre|1977|pp=9}}   In terms of these square representations, the indicator has the following, alternate definition:

:<math displaystyle="block">\iota\chi_V=\begin{cases}
  1 &\text{if }W_{\text{triv}}\text{ is a subrepresentation of }\operatorname{Sym}^2(V) \\
  -1 &\text{if }W_{\text{triv}}\text{ is a subrepresentation of }\operatorname{Alt}^2(V) \\
  0 &\text{otherwise}
\end{cases}</math>

where <math>W_{\text{triv}}</math> is the trivial representation.

To see this, note that the term <math>\chi(g^2)</math> naturally arises in the characters of these representations; to wit, we have<blockquote><math>\chi_V(g^2)=\chi_V(g)^2-2\chi_{\wedge^2V}(g)</math></blockquote>and<blockquote><math>\chi_V(g^2)=2\chi_{\operatorname{Sym}^2(V)}(g)-\chi_V(g)^2</math>.<ref>{{Cite book|title=Representation Theory: A First Course|url=https://archive.org/details/representationth00fult_892|url-access=limited|last=Fulton|first=William|last2=Harris|first2=Joe|publisher=Springer|year=1991|isbn=3-540-97527-6|editor-last=Axler|editor-first=S.|series=Springer Graduate Texts in Mathematics 129|location=New York|pages=[https://archive.org/details/representationth00fult_892/page/n82 13]|author-link=William Fulton (mathematician)|author-link2=Joe Harris (mathematician)|editor-last2=Gehring|editor-first2=F. W.|editor-last3=Ribet|editor-first3=K.|editor-link3=Ken Ribet}}</ref></blockquote>Substituting either of these formulae, the Frobenius–Schur indicator takes on the structure of [[Schur orthogonality relations#Intrinsic statement|the natural {{var|G}}-invariant inner product]] on [[Class function (algebra)|class functions]]:<blockquote><math displaystyle="block">\iota\chi_V =
\begin{cases}
  1 &\langle\chi_{\text{triv}},\chi_{\operatorname{Sym}^2(V)}\rangle=1 \\
  -1 &\langle\chi_{\text{triv}},\chi_{\operatorname{Alt}^2(V)}\rangle=1 \\
  0 &\text{otherwise} \\
\end{cases}</math></blockquote>The inner product counts the multiplicities of [[direct summand]]s; the equivalence of the definitions then follows immediately.

==Applications==

Let {{var|V}} be an irreducible complex representation  of a group {{var|G}} (or equivalently, an irreducible <math>\mathbb{C}[G]</math>-[[Module (mathematics)|module]], where <math>\mathbb{C}[G]</math> denotes the [[group ring]]). Then 
# There exists a nonzero {{var|G}}-invariant [[bilinear form]] on {{var|V}} if and only if <math>\iota\chi\neq 0</math>
# There exists a nonzero {{var|G}}-invariant [[Symmetric bilinear form|''symmetric'' bilinear form]] on {{var|V}} if and only if <math>\iota\chi=1</math>
# There exists a nonzero {{var|G}}-invariant [[Skew-symmetric matrix#Skew-symmetric and alternating forms|''skew-symmetric'' bilinear form]] on {{var|V}} if and only if <math>\iota\chi=-1</math>.{{sfn|James|2001|loc=Theorem 23.16|pp=274}}

The above is a consequence of the [[Universal property|universal properties]] of the [[symmetric algebra]] and [[exterior algebra]], which are the underlying vector spaces of the symmetric and alternating square.

Additionally, 
# <math>\iota\chi=0</math> if and only if <math>\chi</math> is not real-valued (these are complex representations),
# <math>\iota\chi=1</math> if and only if <math>\chi</math> can be realized over <math>\mathbb{R}</math> (these are real representations), and
# <math>\iota\chi=-1</math> if and only if <math>\chi</math> is real but cannot be realized over <math>\mathbb{R}</math> (these are quaternionic representations).{{sfn|James|2001|loc=Corollary 23.17|pp=277}}

==Higher Frobenius-Schur indicators==
Just as for any complex representation ρ,

:<math>\frac{1}{|G|}\sum_{g\in G}\rho(g)</math>

is a self-intertwiner, for any integer ''n'',

:<math>\frac{1}{|G|}\sum_{g\in G}\rho(g^n)</math>

is also a [[intertwiner|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 n<sup>th</sup> ''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 invariant]]s for [[Lie algebra]] irreducible representations. In fact, since any representation of G can be thought of as a [[module (mathematics)|module]] for '''C'''[''G''] and vice versa, we can look at the [[center (algebra)|center]] of '''C'''[''G'']. This is analogous to looking at the center of the [[universal enveloping algebra]] of a Lie algebra. It is simple to check that

:<math>\sum_{g\in G}g^n</math>

belongs to the center of '''C'''[''G''], which is simply the subspace of class functions on ''G''.

==References==
{{Portal|Mathematics}}
{{Refbegin}}
{{Reflist}}
* G.Frobenius & I.Schur, [https://archive.org/details/cbarchive_107811_berdiereellendarstellungendere1882/page/n1/mode/2up Über die reellen Darstellungen der endlichen Gruppen] (1906), Frobenius Gesammelte Abhandlungen vol.III, 354-377.  
*{{cite book|url=https://archive.org/details/linearrepresenta1977serr|first=Jean-Pierre|last=Serre|title=Linear Representations of Finite Groups|publisher=Springer-Verlag|year=1977|isbn=0-387-90190-6|oclc=2202385|url-access=registration}}
* {{Cite book|title=Representations and characters of groups|url=https://archive.org/details/representationsc00jame|url-access=limited|last=James|first=Gordon Douglas|date=2001|publisher=Cambridge University Press|others=[[Liebeck, Martin W]].|isbn=052100392X|edition=2nd|location=Cambridge, UK|oclc=52220683|pp=[https://archive.org/details/representationsc00jame/page/n280 272]–278}}
{{Refend}}

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[[Category:Representation theory of groups]]
[[Category:Representation theory of Lie groups]]