A universal dimension formula for complex simple Lie algebras

Abstract

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 discuss consequences of the formulas including a relation with Scorza varieties.

Dimensional interpolation and the Selberg integral

“…and similar universal rational formulas can be given for the dimensions of irreducible constituents of S 2 g, S 3 g and S 4 g. Although the current status of Vogel's suggestions is unclear to us, these ideas have led to many interesting developments, such as the discovery of E 7 1 2 by Landsberg and Manivel, [LM02], [LM04], [LM06b], [LM06a], [LM06a]. In order to interpolate within the classical A, B, C, D series of Lie algebras, Deligne has defined ⊗-categories…”

X 2 series of universal quantum dimensions

“…As it is shown in [4], the values of scalar product of roots with θ are either 2 (for root θ itself) or 1. These last roots can be organized into three "segments" (see definition below) with unit spacing of (ρ, α) (we normalize the scalar product as in [4] and table 1 by α = −2), which we present below for E 7 as an example: where in the first line there are the values of scalar products with ρ, i.e. the heights ht of roots, in the second -n ht -the number of roots on that height (remember we consider the roots µ with (µ, θ), only).…”

On universal knot polynomials

“…Closer to our purposes in this paper, universal Vogel's [7] and Landsberg and Manivel's [8] formulas (see also [2][3][4][5][6]) deal with (some) irreducible representations in an arbitrary powers of adjoint representation of Lie algebra, while its extension to all representations, including fundamental, is problematic. In this section we collect just some of the relevant technical details.…”

“…One can easily check that dimensions of two sides of (2.37) coincide at an arbitrary values of parameters. Vogel [1,7] and Landsberg and Manivel [8] have found a lot of universal formulas for dimensions of irreps of simple Lie algebras, belonging to powers of adjoint representation. Quantization of most of them is already carried on.…”

“…Actually this is definition of these parameters, which evidently fix them up to common multiplier and permutations. Correspondingly they span the so-called Vogel's plane, which is factor of projective plane over symmetric group, P 2 /S 3 , [8].…”