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

Abstract

If $\mathfrak{g}$ is a complex simple Lie algebra, and $k$ does not exceed the dual Coxeter number of $\mathfrak{g}$, then the k${}^{th}$ coefficient of the $dim\mathfrak{g}$ power of the Euler product may be given by the dimension of a subspace of ${\wedge }^k\mathfrak{g}$ defined by all abelian subalgebras of $\mathfrak{g}$ 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 $\mathfrak{g}$, an element of type $\rho$ and my heat kernel formulation of Macdonald's $\eta$-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 $\mathfrak{g}=LieSl(m,\mathbb{C})$), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac-Moody Lie algebra.

44 pages, plain tex

References in corpus (1)

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra

Cited by in corpus (28)

• Invariant Differential Operators for Non-Compact Lie Algebras Parabolically Related to Conformal Lie Algebras
• Abelian ideals of maximal dimension for solvable Lie algebras
• An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths
• Tree Diagram Lie Algebras of Differential Operators and Evolution Partial Differential Equations
• Conformal blocks, Verlinde formula and diagram automorphisms
• Twisted conformal blocks and their dimension
• On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras, II
• On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces
• Polynomial Representation of $F_4$ and a New Combinatorial Identity about Twenty-Four
• Ad-nilpotent ideals and The Shi arrangement
• Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications
• A Combinatorial Model for Crystals of Kac-Moody Algebras
• Formulas for coefficients of polynomials assigned to arithmetic functions
• A Nekrasov-Okounkov Type formula for $\tilde{C}$
• Affine Weyl groups in K-theory and representation theory
• Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of Z_2-graded Lie algebras
• Casimir operators, abelian subspaces and u-cohomology
• Conjugacy growth series for finitary wreath products
• Lie Algebra Fermions
• Twisted Affine Lie Algebras, Fusion Algebras, and Congruence Subgroups
• Normalisers of abelian ideals of a Borel subalgebra and $\mathbb{Z}$-gradings of a simple Lie algebra
• Symmetries of abelian ideals of Borel subalgebras
• Abelian subalgebras in Z_2-graded Lie algebras and affine Weyl groups
• On Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras
• On 3-Lie algebras with abelian ideals and subalgebras
• Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations
• Conformal embeddings and simple current extensions
• On the Combinatorics of Crystal Graphs, I. Lusztig's Involution