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''to a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259301.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259302.png" /> in a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259303.png" />''
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c0259301.png ~/encyclopedia/old_files/data/C025/C.0205930
 +
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 +
{{TEX|done}}
 +
''to a representation $  \phi $
 +
of a group $  G $
 +
in a linear space $  V $ ''
  
The representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259304.png" /> of the same group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259305.png" /> in the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259306.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259307.png" /> defined by the rule
 
  
<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/c/c025/c025930/c0259308.png" /></td> </tr></table>
+
The representation $  \phi ^{*} $
 +
of the same group $  G $
 +
in the dual space $  V ^{*} $
 +
of $  V $
 +
defined by the rule$$
 +
\phi ^{*} (g)  =
 +
\phi (g ^{-1} ) ^{*}
 +
$$
 +
for all $  g \in G $ ,
 +
where $  * $
 +
denotes taking adjoints.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c0259309.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593010.png" /> denotes taking adjoints.
+
More generally, if $  W $
 +
is a linear space over the same field $  k $
 +
as $  V $
 +
and $  ( \  ,\  ) $
 +
is a non-degenerate [[Bilinear form|bilinear form]] (pairing) on $  V \times W $
 +
with values in $  k $ ,
 +
then a representation $  \psi $
 +
of $  G $
 +
in $  W $
 +
is called the representation contragredient to $  \phi $
 +
with respect to the form $  ( \  ,\  ) $
 +
if$$
 +
( \phi (g) x,\  y)  = 
 +
(x,\  \psi (g ^{-1} ) y)
 +
$$
 +
for all $  g \in G $ ,
 +
$  x \in V $ ,  
 +
$  y \in W $ .
  
More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593011.png" /> is a linear space over the same field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593012.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593014.png" /> is a non-degenerate [[Bilinear form|bilinear form]] (pairing) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593015.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593016.png" />, then a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593019.png" /> is called the representation contragredient to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593020.png" /> with respect to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593021.png" /> if
 
  
<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/c/c025/c025930/c02593022.png" /></td> </tr></table>
+
For example, if $  G $
 +
is the general linear group of a finite-dimensional space $  V $ ,
 +
then the natural representation of $  G $
 +
in the space of covariant tensors of fixed rank on $  V $
 +
is the representation contragredient to the natural representation of $  G $
 +
in the space of contravariant tensors of the same rank on $  V $ .
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593025.png" />.
 
  
For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593026.png" /> is the general linear group of a finite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593027.png" />, then the natural representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593028.png" /> in the space of covariant tensors of fixed rank on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593029.png" /> is the representation contragredient to the natural representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593030.png" /> in the space of contravariant tensors of the same rank on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593031.png" />.
+
Let $  V $
 +
be finite-dimensional over $  k $ ,
 +
let $  (e) $
 +
be a basis of it, and let $  (f \  ) $
 +
be the basis dual to $  (e) $
 +
in $  V ^{*} $ .  
 +
Then, for any $  g $
 +
in $  G $ ,  
 +
the matrix of $  \phi ^{*} (g) $
 +
in the basis $  (f \  ) $
 +
is obtained from the matrix of the operator $  \phi (g) $
 +
in the basis $  (e) $
 +
by taking the transpose of the inverse. If $  \phi $
 +
is irreducible, then so is $  \phi ^{*} $ .  
 +
If $  G $
 +
is a Lie group with Lie algebra $  \mathfrak g $ ,
 +
and $  d \phi $
 +
and $  d \psi $
 +
are the representations of the algebra $  \mathfrak g $
 +
induced, respectively, by two representations $  \phi $
 +
and $  \psi $
 +
of $  G $
 +
in spaces $  V $
 +
and $  W $
 +
that are contragredient with respect to the pairing $  ( \  ,\  ) $ ,
 +
then$$ \tag{*}
 +
(d \phi (X) (x),\  y)  =
 +
- (x,\  d \psi (X) y)
 +
$$
 +
for all $  X \in g $ ,
 +
$  x \in V $ ,
 +
$  y \in W $ .  
 +
Representations of a Lie algebra $  \mathfrak g $
 +
satisfying the condition (*) are also called contragredient representations with respect to $  ( \  ,\  ) $ .
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593032.png" /> be finite-dimensional over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593033.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593034.png" /> be a basis of it, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593035.png" /> be the basis dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593037.png" />. Then, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593039.png" />, the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593040.png" /> in the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593041.png" /> is obtained from the matrix of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593042.png" /> in the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593043.png" /> by taking the transpose of the inverse. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593044.png" /> is irreducible, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593045.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593046.png" /> is a Lie group with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593047.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593049.png" /> are the representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593050.png" /> induced, respectively, by two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593053.png" /> in spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593055.png" /> that are contragredient with respect to the pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593056.png" />, then
 
  
<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/c/c025/c025930/c02593057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
Suppose further that $  G $
 +
is a complex, connected, simply-connected semi-simple Lie group and that $  \phi $
 +
is an irreducible finite-dimensional representation of it in a linear space $  V $ .  
 +
The weights of the representation $  \phi ^{*} $
 +
are opposite to those of $  \phi $ (
 +
see [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]), the lowest weight of $  \phi ^{*} $
 +
being opposite to the highest weight of $  \phi $ (
 +
see [[Cartan theorem|Cartan theorem]] on the highest (weight) vector). The representations $  \phi $
 +
and $  \phi ^{*} $
 +
are equivalent if and only if there is a non-zero bilinear form on $  V $
 +
that is invariant with respect to $  \phi (G) $ .  
 +
If such a form exists, then it is non-degenerate and either symmetric or skew-symmetric. The set of numerical marks of the highest weight of the representation $  \phi ^{*} $
 +
is obtained from the set of numerical marks of $  \phi $
 +
by applying the substitution induced by the following automorphism $  \nu $
 +
of the Dynkin diagram of simple roots $  \Delta $
 +
of $  G $ :
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593060.png" />. Representations of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593061.png" /> satisfying the condition (*) are also called contragredient representations with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593062.png" />.
 
  
Suppose further that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593063.png" /> is a complex, connected, simply-connected semi-simple Lie group and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593064.png" /> is an irreducible finite-dimensional representation of it in a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593065.png" />. The weights of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593066.png" /> are opposite to those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593067.png" /> (see [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]), the lowest weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593068.png" /> being opposite to the highest weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593069.png" /> (see [[Cartan theorem|Cartan theorem]] on the highest (weight) vector). The representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593071.png" /> are equivalent if and only if there is a non-zero bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593072.png" /> that is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593073.png" />. If such a form exists, then it is non-degenerate and either symmetric or skew-symmetric. The set of numerical marks of the highest weight of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593074.png" /> is obtained from the set of numerical marks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593075.png" /> by applying the substitution induced by the following automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593076.png" /> of the Dynkin diagram of simple roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593077.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593078.png" />:
+
a) $  \nu $
 +
takes each connected component $  \Delta _{i} $ ,  
 +
$  i = 1 \dots l $ ,  
 +
of $  \Delta $
 +
into itself;
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593079.png" /> takes each connected component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593081.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593082.png" /> into itself;
+
b) if $  \Delta _{i} $
 
+
is a diagram of type $  A _{r} $ ,  
b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593083.png" /> is a diagram of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593085.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593086.png" />, then the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593087.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593088.png" /> is uniquely defined as the unique element of order 2 in the automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593089.png" />; in the remaining cases the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593090.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593091.png" /> is the identity.
+
$  D _ {2r + 1} $
 +
or $  E _{6} $ ,  
 +
then the restriction of $  \nu $
 +
to $  \Delta _{i} $
 +
is uniquely defined as the unique element of order 2 in the automorphism group of $  \Delta _{i} $ ;  
 +
in the remaining cases the restriction of $  \nu $
 +
to $  \Delta _{i} $
 +
is the identity.
  
 
====References====
 
====References====
Line 33: Line 124:
  
 
====Comments====
 
====Comments====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593092.png" /> is the highest weight of the highest weight representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593093.png" />, then the set of numerical marks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593094.png" /> is simply the ordered set of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025930/c02593096.png" />; cf. [[Cartan theorem|Cartan theorem]], especially when written as labels at the corresponding nodes of the Dynkin diagram.
+
If $  \Lambda \in \mathfrak g ^{*} $
 +
is the highest weight of the highest weight representation $  \phi $ ,  
 +
then the set of numerical marks of $  \Lambda $
 +
is simply the ordered set of integers $  (k _{1} \dots k _{r} ) $ ,  
 +
$  k _{i} = \Lambda (h _{i} ) $ ;  
 +
cf. [[Cartan theorem|Cartan theorem]], especially when written as labels at the corresponding nodes of the Dynkin diagram.

Latest revision as of 10:19, 17 December 2019

c0259301.png ~/encyclopedia/old_files/data/C025/C.0205930 96 0 96 to a representation $ \phi $ of a group $ G $ in a linear space $ V $


The representation $ \phi ^{*} $ of the same group $ G $ in the dual space $ V ^{*} $ of $ V $ defined by the rule$$ \phi ^{*} (g) = \phi (g ^{-1} ) ^{*} $$ for all $ g \in G $ , where $ * $ denotes taking adjoints.

More generally, if $ W $ is a linear space over the same field $ k $ as $ V $ and $ ( \ ,\ ) $ is a non-degenerate bilinear form (pairing) on $ V \times W $ with values in $ k $ , then a representation $ \psi $ of $ G $ in $ W $ is called the representation contragredient to $ \phi $ with respect to the form $ ( \ ,\ ) $ if$$ ( \phi (g) x,\ y) = (x,\ \psi (g ^{-1} ) y) $$ for all $ g \in G $ , $ x \in V $ , $ y \in W $ .


For example, if $ G $ is the general linear group of a finite-dimensional space $ V $ , then the natural representation of $ G $ in the space of covariant tensors of fixed rank on $ V $ is the representation contragredient to the natural representation of $ G $ in the space of contravariant tensors of the same rank on $ V $ .


Let $ V $ be finite-dimensional over $ k $ , let $ (e) $ be a basis of it, and let $ (f \ ) $ be the basis dual to $ (e) $ in $ V ^{*} $ . Then, for any $ g $ in $ G $ , the matrix of $ \phi ^{*} (g) $ in the basis $ (f \ ) $ is obtained from the matrix of the operator $ \phi (g) $ in the basis $ (e) $ by taking the transpose of the inverse. If $ \phi $ is irreducible, then so is $ \phi ^{*} $ . If $ G $ is a Lie group with Lie algebra $ \mathfrak g $ , and $ d \phi $ and $ d \psi $ are the representations of the algebra $ \mathfrak g $ induced, respectively, by two representations $ \phi $ and $ \psi $ of $ G $ in spaces $ V $ and $ W $ that are contragredient with respect to the pairing $ ( \ ,\ ) $ , then$$ \tag{*} (d \phi (X) (x),\ y) = - (x,\ d \psi (X) y) $$ for all $ X \in g $ , $ x \in V $ , $ y \in W $ . Representations of a Lie algebra $ \mathfrak g $ satisfying the condition (*) are also called contragredient representations with respect to $ ( \ ,\ ) $ .


Suppose further that $ G $ is a complex, connected, simply-connected semi-simple Lie group and that $ \phi $ is an irreducible finite-dimensional representation of it in a linear space $ V $ . The weights of the representation $ \phi ^{*} $ are opposite to those of $ \phi $ ( see Weight of a representation of a Lie algebra), the lowest weight of $ \phi ^{*} $ being opposite to the highest weight of $ \phi $ ( see Cartan theorem on the highest (weight) vector). The representations $ \phi $ and $ \phi ^{*} $ are equivalent if and only if there is a non-zero bilinear form on $ V $ that is invariant with respect to $ \phi (G) $ . If such a form exists, then it is non-degenerate and either symmetric or skew-symmetric. The set of numerical marks of the highest weight of the representation $ \phi ^{*} $ is obtained from the set of numerical marks of $ \phi $ by applying the substitution induced by the following automorphism $ \nu $ of the Dynkin diagram of simple roots $ \Delta $ of $ G $ :


a) $ \nu $ takes each connected component $ \Delta _{i} $ , $ i = 1 \dots l $ , of $ \Delta $ into itself;

b) if $ \Delta _{i} $ is a diagram of type $ A _{r} $ , $ D _ {2r + 1} $ or $ E _{6} $ , then the restriction of $ \nu $ to $ \Delta _{i} $ is uniquely defined as the unique element of order 2 in the automorphism group of $ \Delta _{i} $ ; in the remaining cases the restriction of $ \nu $ to $ \Delta _{i} $ is the identity.

References

[1] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001
[3] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
[4] E.B. Vinberg, A.L. Onishchik, "Seminar on algebraic groups and Lie groups 1967/68" , Springer (Forthcoming) (Translated from Russian)


Comments

If $ \Lambda \in \mathfrak g ^{*} $ is the highest weight of the highest weight representation $ \phi $ , then the set of numerical marks of $ \Lambda $ is simply the ordered set of integers $ (k _{1} \dots k _{r} ) $ , $ k _{i} = \Lambda (h _{i} ) $ ; cf. Cartan theorem, especially when written as labels at the corresponding nodes of the Dynkin diagram.

How to Cite This Entry:
Contragredient representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contragredient_representation&oldid=21833
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article