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.