View source for Contragredient representation

c0259301.png ~/encyclopedia/old_files/data/C025/C.0205930
96 0 96
{{TEX|done}}
''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|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|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 $ :


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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)  {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)  {{MR|0412321}} {{ZBL|0342.22001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.P. Zhelobenko,   "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)  {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.B. Vinberg,   A.L. Onishchik,   "Seminar on algebraic groups and Lie groups 1967/68" , Springer  (Forthcoming)  (Translated from Russian)</TD></TR></table>



====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|Cartan theorem]], especially when written as labels at the corresponding nodes of the Dynkin diagram.