Contragredient representation

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.