Dynkin-diagram automorphisms and dual representations

Statement

Let $\mathfrak g$ be a complex semisimple Lie algebra with chosen simple roots $\Delta=\{\alpha_i\}$ . A Dynkin-diagram automorphism is a permutation of $\Delta$ preserving the Cartan matrix. It permutes the fundamental weights and, after choosing a compatible automorphism of $\mathfrak g$ , sends irreducible highest-weight representations to irreducible highest-weight representations with the correspondingly permuted labels.

Let $w_0$ be the longest element of the Weyl group. The opposition involution of the based root system is characterized by

\alpha_i\longmapsto -w_0(\alpha_i).

If $V_\lambda$ is the irreducible representation of highest weight $\lambda$ , then its dual representation has highest weight

\lambda^*=-w_0\lambda.

Thus the opposition involution is precisely the Dynkin-diagram symmetry governing contragredient highest weights.

The simple types

For an irreducible root system, the opposition involution is nontrivial exactly in the following cases:

in type $A_n$ for $n\geq2$ , it reverses the chain, sending $\omega_i$ to $\omega_{n+1-i}$ ;
in type $D_{2k+1}$ , it exchanges the two spin nodes;
in type $E_6$ , it is the unique nontrivial diagram symmetry.

It is the identity in types $A_1$ , $B_n$ , $C_n$ , $D_{2k}$ , $E_7$ , $E_8$ , $F_4$ , and $G_2$ . In those types every finite-dimensional irreducible complex representation is self-dual, although its invariant bilinear form may be symmetric or alternating.

Other diagram symmetries

Not every diagram automorphism is the duality involution. The clearest example is $D_4$ : its full diagram automorphism group is $S_3$ , producing triality, while its opposition involution is the identity. More generally, a diagram symmetry defines a twisting operation on representations whether or not it arises from dualization.

Group-level caution

The highest-weight formula is initially a statement for the complex semisimple Lie algebra, or equivalently for its simply connected complex group. For a non-simply-connected group, only some dominant integral weights exponentiate to representations. Diagram automorphisms that preserve the relevant character lattice descend to that global group; others may exist only on a covering group or on the Lie algebra.

The formula $-w_0\lambda$ is independent of a numbering of diagram nodes. Translating it into named fundamental weights requires a declared Dynkin-labeling convention.

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§13 and 21. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §§13–20. Publisher record.
Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, 2002, Chapters VI, §§1–4. Publisher record.