Branching rule decomposition of the level-1 E₈⁽¹⁾-module with respect to the irregular subalgebra F₄⁽¹⁾ ⊕ G₂⁽¹⁾

2022/06/01 by Carey, Joshua D.
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2206.00163

Abstract

Given a Lie algebra of type E₈, one can use Dynkin diagram automorphisms of the E₆ and D₄ Dynkin diagrams to locate a subalgebra of type F₄⊕ G₂. These automorphisms can be lifted to the affine Kac-Moody counterparts of these algebras and give a subalgebra of type F₄⁽¹⁾⊕ G₂⁽¹⁾ within a type E₈⁽¹⁾ Kac-Moody Lie algebra. We will consider the level-1 irreducible E₈⁽¹⁾-module V^Λ₀ and investigate its branching rule, that is how it decomposes as a direct sum of irreducible F₄⁽¹⁾⊕ G₂⁽¹⁾-modules. We calculate these branching rules using a character formula of Kac-Peterson which uses theta functions and the so-called "string functions." We will make use of Jacobi's, Ramanujan's and the Borweins' theta functions (and their respective properties and identities) in our calculation, including some identities involving the Rogers-Ramanujan series. Virasoro character theory is used to verify string functions stated by Kac and Peterson. We also investigate dissections of some interesting η-quotients.

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