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The Schur index of a central simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834401.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834402.png" /> (cf. [[Central simple algebra|Central simple algebra]]) is the degree of the division algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834403.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834404.png" /> is a full matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834405.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834406.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834407.png" /> be a [[Finite group|finite group]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834408.png" /> a [[Field|field]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s0834409.png" /> the [[Algebraic closure|algebraic closure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344011.png" /> be an irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344012.png" />-module with character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344013.png" /> (cf. [[Irreducible module|Irreducible module]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344014.png" /> be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344015.png" /> by adjoining the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344017.png" />. The Schur index of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344019.png" />, (or the Schur index of the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344020.png" />) is the minimal degree of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344021.png" /> extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344023.png" /> descends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344024.png" />, i.e. such that there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344025.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344027.png" />.
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For a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344028.png" /> the Schur index is always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344029.png" />, [[#References|[a1]]].
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The Schur index of a central simple algebra  $  A $
 +
over a field $  K $(
 +
cf. [[Central simple algebra|Central simple algebra]]) is the degree of the division algebra  $  D $
 +
such that  $  A $
 +
is a full matrix algebra  $  M _ {n} ( D) $
 +
over  $  D $.
  
A basic result on the Schur index is that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344030.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344031.png" /> the multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344033.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344034.png" />.
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Let  $  G $
 +
be a [[Finite group|finite group]],  $  K $
 +
a [[Field|field]] and  $  \overline{K}\; $
 +
the [[Algebraic closure|algebraic closure]] of  $  K $.  
 +
Let  $  V $
 +
be an irreducible  $  \overline{K}\; [ G] $-
 +
module with character  $  \rho $(
 +
cf. [[Irreducible module|Irreducible module]]). Let  $  K( \rho ) $
 +
be obtained from  $  K $
 +
by adjoining the values  $  \rho ( g) $,
 +
$  g \in G $.  
 +
The Schur index of the module  $  V $,
 +
$  m _ {K} ( V) $,
 +
(or the Schur index of the character  $  \rho $)
 +
is the minimal degree of a field  $  S $
 +
extending  $  K( \rho ) $
 +
such that  $  V $
 +
descends to  $  S $,
 +
i.e. such that there is an  $  S[ G] $-
 +
module  $  W $
 +
for which  $  V \simeq \overline{K}\; \otimes _ {S} W $.
  
A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344035.png" /> is a splitting field for a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344036.png" /> if each irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344037.png" />-module is absolutely irreducible, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344038.png" /> is irreducible. The basic result on the Schur index quoted above readily leads to a proof of R. Brauer's result [[#References|[a1]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344039.png" /> is the exponent of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344040.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344041.png" /> is the smallest integer such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344043.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344044.png" /> is a splitting field for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344045.png" />.
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For a finite field $  K $
 +
the Schur index is always  $  1 $,
 +
[[#References|[a1]]].
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344046.png" /> of classes of central simple algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344047.png" /> which occur as components of a group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344048.png" /> for some finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344049.png" /> is a subgroup of the [[Brauer group|Brauer group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344051.png" />, and is known as the Schur subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344052.png" />. Cf. [[#References|[a4]]] for results on the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344053.png" />.
+
A basic result on the Schur index is that for each  $  K[ G] $-
 +
module  $  W $
 +
the multiplicity of  $  V $
 +
in  $  \overline{K}\; \otimes _ {K} W $
 +
is a multiple of  $  m _ {K} ( V) $.
 +
 
 +
A field  $  S \subset  \overline{K}\; $
 +
is a splitting field for a finite group  $  G $
 +
if each irreducible  $  S[ G] $-
 +
module is absolutely irreducible, i.e. if  $  \overline{K}\; \otimes _ {S} V $
 +
is irreducible. The basic result on the Schur index quoted above readily leads to a proof of R. Brauer's result [[#References|[a1]]] that if  $  d $
 +
is the exponent of a finite group  $  G $(
 +
i.e. $  d $
 +
is the smallest integer such that  $  g  ^ {d} = e $
 +
for all  $  g \in G $),
 +
then  $  \mathbf Q ( 1  ^ {1/d} ) $
 +
is a splitting field for  $  G $.
 +
 
 +
The set  $  S( K) $
 +
of classes of central simple algebras over $  K $
 +
which occur as components of a group algebra $  K[ G] $
 +
for some finite group $  G $
 +
is a subgroup of the [[Brauer group|Brauer group]] $  \mathop{\rm Br} ( K) $
 +
of $  K $,  
 +
and is known as the Schur subgroup of $  \mathop{\rm Br} ( K) $.  
 +
Cf. [[#References|[a4]]] for results on the structure of $  S( K) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Brauer,  "On the representation of a group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344054.png" /> in the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344055.png" />-th roots of unity"  ''Amer. J. Math.'' , '''67'''  (1945)  pp. 461–471</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)  pp. §90, §41</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Huppert,  "Finite groups" , '''2''' , Springer  (1982)  pp. §1</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Yamada,  "The Schur subgroup of the Brauer group" , Springer  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Brauer,  "On the representation of a group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344054.png" /> in the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344055.png" />-th roots of unity"  ''Amer. J. Math.'' , '''67'''  (1945)  pp. 461–471</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)  pp. §90, §41</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Huppert,  "Finite groups" , '''2''' , Springer  (1982)  pp. §1</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Yamada,  "The Schur subgroup of the Brauer group" , Springer  (1974)</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


The Schur index of a central simple algebra $ A $ over a field $ K $( cf. Central simple algebra) is the degree of the division algebra $ D $ such that $ A $ is a full matrix algebra $ M _ {n} ( D) $ over $ D $.

Let $ G $ be a finite group, $ K $ a field and $ \overline{K}\; $ the algebraic closure of $ K $. Let $ V $ be an irreducible $ \overline{K}\; [ G] $- module with character $ \rho $( cf. Irreducible module). Let $ K( \rho ) $ be obtained from $ K $ by adjoining the values $ \rho ( g) $, $ g \in G $. The Schur index of the module $ V $, $ m _ {K} ( V) $, (or the Schur index of the character $ \rho $) is the minimal degree of a field $ S $ extending $ K( \rho ) $ such that $ V $ descends to $ S $, i.e. such that there is an $ S[ G] $- module $ W $ for which $ V \simeq \overline{K}\; \otimes _ {S} W $.

For a finite field $ K $ the Schur index is always $ 1 $, [a1].

A basic result on the Schur index is that for each $ K[ G] $- module $ W $ the multiplicity of $ V $ in $ \overline{K}\; \otimes _ {K} W $ is a multiple of $ m _ {K} ( V) $.

A field $ S \subset \overline{K}\; $ is a splitting field for a finite group $ G $ if each irreducible $ S[ G] $- module is absolutely irreducible, i.e. if $ \overline{K}\; \otimes _ {S} V $ is irreducible. The basic result on the Schur index quoted above readily leads to a proof of R. Brauer's result [a1] that if $ d $ is the exponent of a finite group $ G $( i.e. $ d $ is the smallest integer such that $ g ^ {d} = e $ for all $ g \in G $), then $ \mathbf Q ( 1 ^ {1/d} ) $ is a splitting field for $ G $.

The set $ S( K) $ of classes of central simple algebras over $ K $ which occur as components of a group algebra $ K[ G] $ for some finite group $ G $ is a subgroup of the Brauer group $ \mathop{\rm Br} ( K) $ of $ K $, and is known as the Schur subgroup of $ \mathop{\rm Br} ( K) $. Cf. [a4] for results on the structure of $ S( K) $.

References

[a1] R. Brauer, "On the representation of a group of order in the field of -th roots of unity" Amer. J. Math. , 67 (1945) pp. 461–471
[a2] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §90, §41
[a3] B. Huppert, "Finite groups" , 2 , Springer (1982) pp. §1
[a4] T. Yamada, "The Schur subgroup of the Brauer group" , Springer (1974)
How to Cite This Entry:
Schur index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_index&oldid=48623