Weyl's character formula

In the mathematics which is Weyl character formula or character of formula Weyl a formula for calculating the character of a representation of their highest weight.

It was proved in 1926 by Hermann Weyl and also follows from the Atiyah-Bott fixed point theorem .

background

Let be a compact Lie group and a maximal torus. A weight of a representation is a mapping for which there are vectors with for all . ${\displaystyle G}$ ${\displaystyle T\subset G}$ ${\displaystyle \rho \colon G\to GL(n,\mathbb {C} )}$ ${\displaystyle \lambda \colon T\to C^{*}}$ ${\displaystyle v\not =0}$ ${\displaystyle \rho (h)v=\lambda (h)v}$ ${\displaystyle h\in T}$

The choice of a Weyl chamber or, equivalently, a system of positive roots gives a partial order on the weights, in particular one can speak of the highest weight of a representation. The theorem of greatest weight says that for everyone with there is a unique irreducible representation with the highest weight , and that every irreducible representation can be obtained in this way. ${\displaystyle \Delta ^{+}}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda (\Delta ^{+})\subset \mathbb {N} }$ ${\displaystyle \lambda }$

In particular, the characters of a representation depend only on their highest weight. Weyl's character formula gives an explicit description of this connection.

formula

Let be a compact connected Lie group and a maximal torus . For a root system of be the positive roots and the Weyl group . ${\displaystyle G}$ ${\displaystyle T\subset G}$ ${\displaystyle \Delta }$ ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }}$ ${\displaystyle \Delta ^{+}}$ ${\displaystyle W=\left\{r_{\alpha }\colon \alpha \in \Delta \right\}}$

Then applies to the character of an irreducible representation with the highest weight ${\displaystyle \chi _{\lambda }}$ ${\displaystyle \lambda }$

${\displaystyle \chi _{\lambda }(g)=\Theta _{\lambda }(g)}$

for everyone taking the through ${\displaystyle g\in G}$ ${\displaystyle \Theta _{\lambda }}$

${\displaystyle \Theta _{\lambda }(H)={\frac {\sum _{w\in W}\det(w)e^{w(\lambda +\rho )H}}{\Pi _{\alpha \in \Delta ^{+}}(e^{\alpha (H)/2}-e^{-\alpha (H)/2})}}}$

for all with a clearly defined smooth class function . ${\displaystyle H\in {\mathfrak {t}}}$ ${\displaystyle \alpha (H)\not \in 2\pi i\mathbb {Z} \ \forall \alpha \in \Delta }$ ${\displaystyle \Theta _{\lambda }\colon G^{reg}\to \mathbb {C} }$

literature

• H. Weyl: Theory of the representation of continuous semi-simple groups by linear transformations. III, Mathematische Zeitschrift 24 (1926), 377-395.
• MF Atiyah, R. Bott: A Lefschetz fixed point theorem for elliptic complexes: II. Applications, Annals of Mathematics 88 (1968), 451-491.

Web links

• B. Sury: Hermann Weyl and Representation Theory