Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra

Abstract

ABSTRACT. If g is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of g, then the k th coefficient of the dim g power of the Euler product may be given by the dimension of a subspace of ∧ k g defined by all abelian subalgebras of g of dimension k.This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Peterson's 2 rank theorem on the number of abelian ideals in a Borel subalgebra of g, an element of type ρ and my heat kernel formulation of Macdonald's η-function theorem, a set D alcove of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null m-core when g = Lie Sl(m, C)), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac-Moody Lie algebra.

On the combinatorics of crystal graphs, I. Lusztig's involution

“…Each alcove A contains precisely one element ζ A of the set Z (cf. [15,22]); this will be called the central point of A. In particular, ζ A • = ρ/ h.…”

Partial differential equation approach to F 4

“…(1.1) Kostant [7] found a connection of the above identity with the abelian subalgebras of G . Let λ r be the rth fundamental weight of G .…”

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

“…On the other hand, by [Ko2,Theorem 4.5 and Remark 4.13], for w ∈ Aff 2 (W ), V (w) ⊂ Im(i). This proves the surjectivity of ξ and hence concludes the proof that ξ is a graded algebra isomorphism.…”

“…Recall that there is a bijection ζ : Ξ → Aff 2 (W ) such that for any I ∈ Ξ, dim I = (ζ(I)) (see, e.g., [Ko2,Theorem 4.4] [S,Proposition 11], for any I ∈ Ξ o , dim I ≤ h − 1. In particular, this gives dim C Z ≤ h − 1.…”