# Character formula

Weyl formula

A formula that expresses the character $\mathop{\rm ch} V ( \Lambda )$ of an irreducible finite-dimensional representation of a semi-simple Lie algebra $\mathfrak g$ over an algebraically closed field of characteristic 0 (cf. Character of a finite-dimensional representation of a semi-simple Lie algebra) in terms of its highest weight $\Lambda$:

$$\mathop{\rm ch} V ( \Lambda ) = \ \frac{\sum _ {w \in W } ( \mathop{\rm det} w) e ^ {w ( \Lambda + \rho ) } }{\sum _ {w \in W } ( \mathop{\rm det} w) e ^ {w ( \rho ) } } =$$

$$= \ \frac{\sum _ {w \in W } ( \mathop{\rm det} w) e ^ {w ( \Lambda + \rho ) - \rho } }{\prod _ {\alpha \in \mathbf R ^ {+} } (1 - e ^ {- \alpha } ) }$$

(here $W$ is the Weyl group and $\rho = ( \sum _ {\alpha \in \mathbf R ^ {+} } \alpha )/2$ is half the sum of the positive roots of the Lie algebra $\mathfrak g$). Consequences of the character formula are the formula for the dimension of the representation:

$$\mathop{\rm dim} V ( \Lambda ) = \ \prod _ {\alpha \in \mathbf R ^ {+} } \frac{( \Lambda + \rho , \alpha ) }{( \rho , \alpha ) } ,$$

a formula for the multiplicity of a weight, and also Steinberg's formula for the number $m _ \Lambda$ of occurrences of the irreducible $\mathfrak g$- module $V ( \Lambda )$ in $V ( \Lambda ^ \prime ) \otimes V ( \Lambda ^ {\prime\prime} )$:

$$m _ \Lambda = \ \sum _ {s, t \in W } \mathop{\rm det} (st) P ( \Lambda + 2 \rho - s ( \Lambda ^ \prime + \rho ) - t ( \Lambda ^ {\prime\prime} + \rho )),$$

where $P ( \mu )$ is the number of distinct presentations of an element $\mu$ as a sum of positive roots (see ).

The character formula can be generalized to the case of irreducible representations of graded Lie algebras defined by an indecomposable Cartan matrix (see also Lie algebra, graded). This generalization leads to the following combinatorial identities:

$$e (t) = \ \sum _ {j \in \mathbf Z } (-1) ^ {j} t ^ {j (3j + 1)/2 }$$

(Euler's identity);

$$\frac{e ^ {2} (t) }{e (t ^ {2} ) } = \ \sum _ {j \in \mathbf Z } (-1) ^ {j} t ^ {j ^ {2} }$$

(Gauss' identity);

$$e ^ {3} (t) = \ \sum _ {j \geq 0 } (-1) ^ {j} (2j + 1) ^ {j (j + 1)/2 }$$

(Jacobi's identity); where

$$e (t) = \ \prod _ {n \geq 1 } (1 - t ^ {n} );$$

$$\prod _ {j \geq 1 } (1 - t ^ {5j - 4 } ) ^ {-1} (1 - t ^ {5j - 1 } ) ^ {-1} = \sum _ {n \geq 0 } \frac{t ^ {n ^ {2} } }{(1 - t) \dots (1 - t ^ {n} ) }$$

(the Rogers–Ramanujan identity); and others (see , ).

An analogue of the character formula can also be obtained for irreducible representations of certain simple Lie superalgebras (cf. Superalgebra) .

How to Cite This Entry:
Character formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_formula&oldid=44945
This article was adapted from an original article by D.A. Leites (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article