In mathematics, the Chang number of an irreducible representation of a simple complex

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

is its dimension modulo 1 + ''h'', where ''h'' is the

Coxeter number In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

. 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 group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

s ''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.

References

* *{{Citation , authorlink=Victor Kac , last1=Kac , first1=Victor G , title=Algebra, Carbondale 1980 (Proc. Conf., Southern Illinois Univ., Carbondale, Ill., 1980) , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

, location=Berlin, New York , series=Lecture Notes in Math. , doi=10.1007/BFb0090559 , mr=613179 , year=1981 , volume=848 , chapter=Simple Lie groups and the Legendre symbol , pages=110–123, isbn=978-3-540-10573-2 Representation theory