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arxiv math/0401296 v2 pith:L5MHPGIN submitted 2004-01-22 math.RT math.AG

A universal dimension formula for complex simple Lie algebras

classification math.RT math.AG
keywords formulasuniversalcomplexformuladimensionfunctionssimpleadjoint
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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.

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  1. Diagrammatic technique for Vogel's universality

    math.QA 2026-05 unverdicted novelty 5.0

    Vogel's diagrammatic Lambda-algebra enables truly universal computations of Lie-theoretic quantities, demonstrated via multiple examples.