Chang number

In mathematics, the Chang number of an irreducible representation of a simple complex Lie algebra is its dimension modulo 1 + "h", where "h" is the Coxeter number. Chang numbers are named after , who rediscovered an element of order "h" + 1 found by .
 showed that there is a unique class of regular elements σ of order "h" + 1, in the complex points of the corresponding Chevalley group. He showed that the trace of σ on an irreducible representation is −1, 0, or +1, and if "h" + 1 is prime then the trace is congruent to the dimension mod "h"+1. This implies that the dimension of an irreducible representation is always −1, 0, or +1 mod "h" + 1 whenever "h" + 1 is prime.
Examples.
In particular, for the exceptional compact Lie groups "G"2, F4, E6, E7, and E8 the number "h" + 1 = 7, 13, 13, 19, 31 is always prime, so the Chang number of an irreducible representation is always +1, 0, or −1.
For example, the first few irreducible representations of G2 (with Coxeter number "h" = 6) have dimensions 1, 7, 14, 27, 64, 77, 182, 189, 273, 286...
These are congruent to 1, 0, 0, −1, 1, 0, 0, 0, 0, −1... mod 7 = "h" + 1.