A universal dimension formula for complex simple Lie algebras

Joseph M. Landsberg (1) , Laurent Manivel (2)

1 TAMU - Department of Mathematics [Texas]
2 IF - Institut Fourier

We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as several other universal formulas. These formulas generalize formulas of Vogel and Deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients. We also discuss some consequences of the formulas including a relation with Scorza varieties.

Mots clés: simple Lie algebra; adjoint representation; Hilbert function; Scorza variety

Domaines: Géométrie algébrique [math.AG]

Joseph M. Landsberg, Laurent Manivel. A universal dimension formula for complex simple Lie algebras. Advances in Mathematics, 2006, 201 (2), pp.379-407. ⟨hal-00330641⟩