# Contragredient representation

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*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=44269