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''of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776301.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776302.png" />''
+
{{TEX|done}}
  
A [[Linear representation|linear representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776303.png" /> on a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776304.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776305.png" /> which is a rational homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776306.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776307.png" />. One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r0776308.png" /> is a rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763010.png" />-module. Direct sums and tensor products of a finite number of rational representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763011.png" /> are rational representations. Subrepresentations and quotient representations of any rational representation are rational representations. Symmetric and exterior powers of any rational representation are rational representations. The representation contragredient to a rational representation is a rational representation.
+
''of an algebraic group  $  G $
 +
over an algebraically closed field  $  k $''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763012.png" /> is finite, then each of its linear representations will be a rational representation, and the theory of rational representations coincides with the theory of representations of finite groups (cf. [[Representation of a group|Representation of a group]]). To a large extent, specific methods of the theory of linear algebraic groups are used to study rational representations in case the group under consideration is connected, and the most thoroughly developed theory is that of rational representations of connected semi-simple algebraic groups. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763013.png" /> be such a group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763014.png" /> a [[Maximal torus|maximal torus]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763015.png" /> its group of rational characters (written additively), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763016.png" /> the [[Root system|root system]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763017.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763019.png" /> its [[Weyl group|Weyl group]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763020.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763021.png" />-invariant positive-definite non-degenerate scalar product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763022.png" />. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763023.png" /> be a rational representation. The restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763024.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763025.png" /> decomposes into a direct sum of one-dimensional representations; more precisely,
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763026.png" /></td> </tr></table>
+
A [[Linear representation|linear representation]] of  $  G $
 +
on a finite-dimensional vector space  $  V $
 +
over  $  k $
 +
which is a rational homomorphism of  $  G $
 +
into  $  \mathop{\rm GL}\nolimits (V) $.  
 +
One also says that  $  V $
 +
is a rational  $  G $-
 +
module. Direct sums and tensor products of a finite number of rational representations of  $  G $
 +
are rational representations. Subrepresentations and quotient representations of any rational representation are rational representations. Symmetric and exterior powers of any rational representation are rational representations. The representation contragredient to a rational representation is a rational representation.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763027.png" /> is some set of characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763028.png" />, called the weights of the representation, and
+
If  $  G $
 +
is finite, then each of its linear representations will be a rational representation, and the theory of rational representations coincides with the theory of representations of finite groups (cf. [[Representation of a group|Representation of a group]]). To a large extent, specific methods of the theory of linear algebraic groups are used to study rational representations in case the group under consideration is connected, and the most thoroughly developed theory is that of rational representations of connected semi-simple algebraic groups. Let  $  G $
 +
be such a group,  $  T $
 +
a [[Maximal torus|maximal torus]],  $  X (T) $
 +
its group of rational characters (written additively),  $  \Sigma $
 +
the [[Root system|root system]] of  $  G $
 +
with respect to  $  T $,
 +
$  W $
 +
its [[Weyl group|Weyl group]], and  $  ( \  ,\  ) $
 +
a  $  W $-
 +
invariant positive-definite non-degenerate scalar product on  $  X(T) \otimes \mathbf R $.  
 +
Now let  $  \phi : \  G \rightarrow  \mathop{\rm GL}\nolimits (V) $
 +
be a rational representation. The restriction of  $  \phi $
 +
to  $  T $
 +
decomposes into a direct sum of one-dimensional representations; more precisely, $$
 +
V  =  \oplus _ {\chi \in P _ \phi } V ( \chi ) ,
 +
$$
 +
where  $  P _ \phi  \subset X (T) $
 +
is some set of characters of $  T $,  
 +
called the weights of the representation, and $$
 +
V ( \chi )  =  \{ {v \in V} : {\phi (t) v = \chi (t) v  \forall t \in T} \}
 +
  \neq  0 .
 +
$$
 +
The set of weights  $  P _ \phi  $
 +
is invariant under the action of  $  W $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763029.png" /></td> </tr></table>
 
  
The set of weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763030.png" /> is invariant under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763031.png" />.
+
If  $  \mathop{\rm char}\nolimits \  k = 0 $,
 +
then every rational representation of  $  G $
 +
is completely reducible, but if  $  \mathop{\rm char}\nolimits \  k > 0 $,
 +
then this is not so (see [[Mumford hypothesis|Mumford hypothesis]]). Whatever the characteristic of $  k $,
 +
however, there is a complete description of the irreducible rational representations.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763032.png" />, then every rational representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763033.png" /> is completely reducible, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763034.png" />, then this is not so (see [[Mumford hypothesis|Mumford hypothesis]]). Whatever the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763035.png" />, however, there is a complete description of the irreducible rational representations.
+
Let  $  B $
 +
be a [[Borel subgroup|Borel subgroup]] in  $  G $
 +
containing  $  T $
 +
and let  $  \Delta $
 +
be the set of simple roots in  $  \Sigma $
 +
defined by  $  B $.  
 +
Identify the group  $  X(B) $
 +
of rational characters of  $  B $
 +
with  $  X (T) $.  
 +
In the space  $  V $,  
 +
for any irreducible rational representation $  \phi : \  G \rightarrow  \mathop{\rm GL}\nolimits (V) $
 +
there is a unique one-dimensional weight subspace  $  V ( \delta _ \phi  ) $,
 +
$  \delta _ \phi  \in P _ \phi  $,
 +
invariant under  $  B $.  
 +
The character  $  \delta _ \phi  $
 +
is called the highest weight of the irreducible rational representation  $  \phi $;
 +
it is dominant, i.e. $  ( \delta _ \phi  ,\  \alpha ) \geq 0 $
 +
for any  $  \alpha \in \Delta $,  
 +
and every other weight  $  \chi \in P _ \phi  $
 +
has the form $$
 +
\chi  =   \delta _ \phi  - \sum _ {\alpha \in \Delta}
 +
m _ \alpha  \alpha , 
 +
m _ \alpha  \in \mathbf Z , 
 +
m _ \alpha  \geq 0 .
 +
$$
 +
The mapping  $  \phi \mapsto \delta _ \phi  $
 +
defines a bijection between the classes of equivalent irreducible rational representations and the dominant elements of  $  X (T) $.
 +
An explicit construction of all irreducible rational representations can be obtained in the following way. Let  $  k [ G ] $
 +
be the algebra of regular functions on  $  G $.  
 +
Given any  $  \chi \in X (T) = X (B) $,  
 +
consider the subspace $$
 +
k [ G ] _ \chi    = 
 +
\{ {f \in k [ G ]} : {f ( g b ) = \chi (b) f (g)   \forall b \in B ,
 +
g \in G} \}
 +
.
 +
$$
 +
It is finite-dimensional and is a rational  $  G $-
 +
module under the action of  $  G $
 +
by left translation. The geometric meaning of this space is as follows: it can be canonically identified with the set of regular sections of the one-dimensional homogeneous vector bundle over  $  G / B $
 +
determined by the character  $  - \chi $.  
 +
Let  $  w _{0} \in W $
 +
be the element mapping positive roots into negative ones. If  $  k [ G ] _{ {-} w _{0} ( \chi )} \neq 0 $,  
 +
then  $  \chi $
 +
is a dominant character and the minimal non-zero  $  G $-
 +
submodule in  $  k [ G ] _{ {-} w _{0} ( \chi )} $
 +
is an irreducible rational  $  G $-
 +
module with highest weight  $  \chi $.
 +
Every irreducible rational $  G $-
 +
module can be obtained in this way. If  $  \mathop{\rm char}\nolimits \  k = 0 $,
 +
then the  $  G $-
 +
module  $  k [ G ] _{ {-} w _{0} ( \chi )} $
 +
is itself irreducible.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763036.png" /> be a [[Borel subgroup|Borel subgroup]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763037.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763038.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763039.png" /> be the set of simple roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763040.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763041.png" />. Identify the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763042.png" /> of rational characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763043.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763044.png" />. In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763045.png" />, for any irreducible rational representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763046.png" /> there is a unique one-dimensional weight subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763048.png" />, invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763049.png" />. The character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763050.png" /> is called the highest weight of the irreducible rational representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763051.png" />; it is dominant, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763052.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763053.png" />, and every other weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763054.png" /> has the form
+
To obtain irreducible rational representations, one often applies the above-mentioned operations to given rational representations. For example, if  $  \phi _{i} $
 +
is an irreducible rational representation with highest weight $  \chi _{i} $,  
 +
$  i = 1 \dots d $,  
 +
then some quotient representation of  $  \phi _{1} \otimes \dots \otimes \phi _{d} $
 +
is an irreducible rational representation with highest weight  $  \chi _{1} + \dots + \chi _{d} $(
 +
it is called the Cartan product of  $  \phi _{1} \dots \phi _{d} $).
 +
If  $  \phi $
 +
is an irreducible rational representation with highest weight $  \chi $,
 +
then some quotient representation of $  S _ \phi  ^{d} $
 +
is an irreducible rational representation with highest weight  $  d \chi $.  
 +
Moreover  $  \phi ^{*} $
 +
is irreducible and its highest weight is  $  - w _{0} ( \chi ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763055.png" /></td> </tr></table>
 
  
The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763056.png" /> defines a bijection between the classes of equivalent irreducible rational representations and the dominant elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763057.png" />. An explicit construction of all irreducible rational representations can be obtained in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763058.png" /> be the algebra of regular functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763059.png" />. Given any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763060.png" />, consider the subspace
+
Let  $  \mathfrak g $
 +
be the Lie algebra of  $  G $(
 +
cf. [[Lie algebra of an algebraic group|Lie algebra of an algebraic group]]). If  $  \phi : \  G \rightarrow  \mathop{\rm GL}\nolimits (V) $
 +
is a rational representation, then its differential  $  d \phi $
 +
is a representation of the Lie algebra  $  \mathfrak g $.
 +
A rational representation  $  \phi $
 +
is called infinitesimally irreducible if  $  d \phi $
 +
is an irreducible representation of the algebra  $  \mathfrak g $.
 +
An infinitesimally-irreducible rational representation is irreducible, and when  $  \mathop{\rm char}\nolimits \  k = 0 $,
 +
the converse is also true (which largely reduces the theory of rational representations of a group to the theory of representations of its Lie algebra). But when  $  \mathop{\rm char}\nolimits \  k =p > 0 $,  
 +
this is not so; the infinitesimally-irreducible rational representations in this case are just those irreducible rational representations with highest weight  $  \chi $
 +
for which $$
 +
0  \leq 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763061.png" /></td> </tr></table>
+
\frac{2 ( \chi ,\  \alpha )}{( \alpha ,\  \alpha )}
  
It is finite-dimensional and is a rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763062.png" />-module under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763063.png" /> by left translation. The geometric meaning of this space is as follows: it can be canonically identified with the set of regular sections of the one-dimensional homogeneous vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763064.png" /> determined by the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763065.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763066.png" /> be the element mapping positive roots into negative ones. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763067.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763068.png" /> is a dominant character and the minimal non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763069.png" />-submodule in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763070.png" /> is an irreducible rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763071.png" />-module with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763072.png" />. Every irreducible rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763073.png" />-module can be obtained in this way. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763074.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763075.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763076.png" /> is itself irreducible.
+
  <   p    \textrm{ for  all } 
 +
\alpha \in \Delta .
 +
$$
 +
Moreover, all the irreducible rational representations can be constructed using the infinitesimally-irreducible ones. More precisely, if  $  G $
 +
is simply connected, that is, if  $  X (T) $
 +
coincides with the lattice of weights of the root system  $  \Sigma $,  
 +
then every irreducible rational representation factors uniquely into a tensor product of the form $$
 +
\phi _{0} \otimes
 +
\phi _{1} ^{ \textrm  Fr} \otimes \dots
 +
\otimes \phi _{d} ^{ \textrm  Fr ^{d}} ,
 +
$$
 +
where  $  \phi _{0} \dots \phi _{d} $
 +
are infinitesimally irreducible, and  $  \phi _{i} ^{ \textrm  Fr ^{i}} $
 +
is the representation obtained by applying the Frobenius automorphism  $  a \mapsto a ^{ {p} ^{i}} $(
 +
$  a \in k $,
 +
$  p = \mathop{\rm char}\nolimits \  k $)
 +
to the matrix entries of the representation  $  \phi _{i} $.
  
To obtain irreducible rational representations, one often applies the above-mentioned operations to given rational representations. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763077.png" /> is an irreducible rational representation with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763079.png" />, then some quotient representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763080.png" /> is an irreducible rational representation with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763081.png" /> (it is called the Cartan product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763082.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763083.png" /> is an irreducible rational representation with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763084.png" />, then some quotient representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763085.png" /> is an irreducible rational representation with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763086.png" />. Moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763087.png" /> is irreducible and its highest weight is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763088.png" />.
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763089.png" /> be the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763090.png" /> (cf. [[Lie algebra of an algebraic group|Lie algebra of an algebraic group]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763091.png" /> is a rational representation, then its differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763092.png" /> is a representation of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763093.png" />. A rational representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763094.png" /> is called infinitesimally irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763095.png" /> is an irreducible representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763096.png" />. An infinitesimally-irreducible rational representation is irreducible, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763097.png" />, the converse is also true (which largely reduces the theory of rational representations of a group to the theory of representations of its Lie algebra). But when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763098.png" />, this is not so; the infinitesimally-irreducible rational representations in this case are just those irreducible rational representations with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763099.png" /> for which
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630100.png" /></td> </tr></table>
 
 
Moreover, all the irreducible rational representations can be constructed using the infinitesimally-irreducible ones. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630101.png" /> is simply connected, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630102.png" /> coincides with the lattice of weights of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630103.png" />, then every irreducible rational representation factors uniquely into a tensor product of the form
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630104.png" /></td> </tr></table>
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630105.png" /> are infinitesimally irreducible, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630106.png" /> is the representation obtained by applying the Frobenius automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630107.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630109.png" />) to the matrix entries of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630110.png" />.
 
  
 
====References====
 
====References====
Line 43: Line 156:
  
 
====Comments====
 
====Comments====
See [[#References|[a1]]], especially for complications in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630111.png" />.
+
See [[#References|[a1]]], especially for complications in $  \mathop{\rm char}\nolimits \  k > 0 $.
 +
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) {{MR|0899071}} {{ZBL|0654.20039}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) {{MR|0899071}} {{ZBL|0654.20039}} </TD></TR></table>

Latest revision as of 11:57, 20 December 2019


of an algebraic group $ G $ over an algebraically closed field $ k $


A linear representation of $ G $ on a finite-dimensional vector space $ V $ over $ k $ which is a rational homomorphism of $ G $ into $ \mathop{\rm GL}\nolimits (V) $. One also says that $ V $ is a rational $ G $- module. Direct sums and tensor products of a finite number of rational representations of $ G $ are rational representations. Subrepresentations and quotient representations of any rational representation are rational representations. Symmetric and exterior powers of any rational representation are rational representations. The representation contragredient to a rational representation is a rational representation.

If $ G $ is finite, then each of its linear representations will be a rational representation, and the theory of rational representations coincides with the theory of representations of finite groups (cf. Representation of a group). To a large extent, specific methods of the theory of linear algebraic groups are used to study rational representations in case the group under consideration is connected, and the most thoroughly developed theory is that of rational representations of connected semi-simple algebraic groups. Let $ G $ be such a group, $ T $ a maximal torus, $ X (T) $ its group of rational characters (written additively), $ \Sigma $ the root system of $ G $ with respect to $ T $, $ W $ its Weyl group, and $ ( \ ,\ ) $ a $ W $- invariant positive-definite non-degenerate scalar product on $ X(T) \otimes \mathbf R $. Now let $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ be a rational representation. The restriction of $ \phi $ to $ T $ decomposes into a direct sum of one-dimensional representations; more precisely, $$ V = \oplus _ {\chi \in P _ \phi } V ( \chi ) , $$ where $ P _ \phi \subset X (T) $ is some set of characters of $ T $, called the weights of the representation, and $$ V ( \chi ) = \{ {v \in V} : {\phi (t) v = \chi (t) v \forall t \in T} \} \neq 0 . $$ The set of weights $ P _ \phi $ is invariant under the action of $ W $.


If $ \mathop{\rm char}\nolimits \ k = 0 $, then every rational representation of $ G $ is completely reducible, but if $ \mathop{\rm char}\nolimits \ k > 0 $, then this is not so (see Mumford hypothesis). Whatever the characteristic of $ k $, however, there is a complete description of the irreducible rational representations.

Let $ B $ be a Borel subgroup in $ G $ containing $ T $ and let $ \Delta $ be the set of simple roots in $ \Sigma $ defined by $ B $. Identify the group $ X(B) $ of rational characters of $ B $ with $ X (T) $. In the space $ V $, for any irreducible rational representation $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ there is a unique one-dimensional weight subspace $ V ( \delta _ \phi ) $, $ \delta _ \phi \in P _ \phi $, invariant under $ B $. The character $ \delta _ \phi $ is called the highest weight of the irreducible rational representation $ \phi $; it is dominant, i.e. $ ( \delta _ \phi ,\ \alpha ) \geq 0 $ for any $ \alpha \in \Delta $, and every other weight $ \chi \in P _ \phi $ has the form $$ \chi = \delta _ \phi - \sum _ {\alpha \in \Delta} m _ \alpha \alpha , m _ \alpha \in \mathbf Z , m _ \alpha \geq 0 . $$ The mapping $ \phi \mapsto \delta _ \phi $ defines a bijection between the classes of equivalent irreducible rational representations and the dominant elements of $ X (T) $. An explicit construction of all irreducible rational representations can be obtained in the following way. Let $ k [ G ] $ be the algebra of regular functions on $ G $. Given any $ \chi \in X (T) = X (B) $, consider the subspace $$ k [ G ] _ \chi = \{ {f \in k [ G ]} : {f ( g b ) = \chi (b) f (g) \forall b \in B , g \in G} \} . $$ It is finite-dimensional and is a rational $ G $- module under the action of $ G $ by left translation. The geometric meaning of this space is as follows: it can be canonically identified with the set of regular sections of the one-dimensional homogeneous vector bundle over $ G / B $ determined by the character $ - \chi $. Let $ w _{0} \in W $ be the element mapping positive roots into negative ones. If $ k [ G ] _{ {-} w _{0} ( \chi )} \neq 0 $, then $ \chi $ is a dominant character and the minimal non-zero $ G $- submodule in $ k [ G ] _{ {-} w _{0} ( \chi )} $ is an irreducible rational $ G $- module with highest weight $ \chi $. Every irreducible rational $ G $- module can be obtained in this way. If $ \mathop{\rm char}\nolimits \ k = 0 $, then the $ G $- module $ k [ G ] _{ {-} w _{0} ( \chi )} $ is itself irreducible.

To obtain irreducible rational representations, one often applies the above-mentioned operations to given rational representations. For example, if $ \phi _{i} $ is an irreducible rational representation with highest weight $ \chi _{i} $, $ i = 1 \dots d $, then some quotient representation of $ \phi _{1} \otimes \dots \otimes \phi _{d} $ is an irreducible rational representation with highest weight $ \chi _{1} + \dots + \chi _{d} $( it is called the Cartan product of $ \phi _{1} \dots \phi _{d} $). If $ \phi $ is an irreducible rational representation with highest weight $ \chi $, then some quotient representation of $ S _ \phi ^{d} $ is an irreducible rational representation with highest weight $ d \chi $. Moreover $ \phi ^{*} $ is irreducible and its highest weight is $ - w _{0} ( \chi ) $.


Let $ \mathfrak g $ be the Lie algebra of $ G $( cf. Lie algebra of an algebraic group). If $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ is a rational representation, then its differential $ d \phi $ is a representation of the Lie algebra $ \mathfrak g $. A rational representation $ \phi $ is called infinitesimally irreducible if $ d \phi $ is an irreducible representation of the algebra $ \mathfrak g $. An infinitesimally-irreducible rational representation is irreducible, and when $ \mathop{\rm char}\nolimits \ k = 0 $, the converse is also true (which largely reduces the theory of rational representations of a group to the theory of representations of its Lie algebra). But when $ \mathop{\rm char}\nolimits \ k =p > 0 $, this is not so; the infinitesimally-irreducible rational representations in this case are just those irreducible rational representations with highest weight $ \chi $ for which $$ 0 \leq \frac{2 ( \chi ,\ \alpha )}{( \alpha ,\ \alpha )} < p \textrm{ for all } \alpha \in \Delta . $$ Moreover, all the irreducible rational representations can be constructed using the infinitesimally-irreducible ones. More precisely, if $ G $ is simply connected, that is, if $ X (T) $ coincides with the lattice of weights of the root system $ \Sigma $, then every irreducible rational representation factors uniquely into a tensor product of the form $$ \phi _{0} \otimes \phi _{1} ^{ \textrm Fr} \otimes \dots \otimes \phi _{d} ^{ \textrm Fr ^{d}} , $$ where $ \phi _{0} \dots \phi _{d} $ are infinitesimally irreducible, and $ \phi _{i} ^{ \textrm Fr ^{i}} $ is the representation obtained by applying the Frobenius automorphism $ a \mapsto a ^{ {p} ^{i}} $( $ a \in k $, $ p = \mathop{\rm char}\nolimits \ k $) to the matrix entries of the representation $ \phi _{i} $.


References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] A. Borel, "Linear representations of semi-simple algebraic groups" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 421–440 MR0372054 Zbl 0311.20022
[3] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
[4] A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , Seminar on algebraic groups and related finite groups , Lect. notes in math. , 131 , Springer (1970) Zbl 0192.36201
[5] R.G. Steinberg, "Lectures on Chevalley groups" , Yale Univ. Press (1968) MR0466335 Zbl 1196.22001
[6] R. Steinberg, "Representations of algebraic groups" Nagoya Math. J. , 22 (1963) pp. 33–56 MR0155937 Zbl 0271.20019
[7] G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) MR0207883 Zbl 0131.02702
[8] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) MR0323842 Zbl 0254.17004


Comments

See [a1], especially for complications in $ \mathop{\rm char}\nolimits \ k > 0 $.


References

[a1] J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) MR0899071 Zbl 0654.20039
How to Cite This Entry:
Rational representation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Rational_representation&oldid=44312
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article