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Branching rule decomposition of the level-1 $E_8^{(1)}$-module with respect to the irregular subalgebra $F_4^{(1)} oplus G_2^{(1)}$ | | Joshua D. Carey
; | | Date: |
1 Jun 2022 | | Abstract: | Given a Lie algebra of type $E_8$, one can use Dynkin diagram automorphisms
of the $E_6$ and $D_4$ Dynkin diagrams to locate a subalgebra of type
$F_4oplus G_2$. These automorphisms can be lifted to the affine Kac-Moody
counterparts of these algebras and give a subalgebra of type $F_4^{(1)}oplus
G_2^{(1)}$ within a type $E_8^{(1)}$ Kac-Moody Lie algebra. We will consider
the level-1 irreducible $E_8^{(1)}$-module $V^{Lambda_0}$ and investigate its
branching rule, that is how it decomposes as a direct sum of irreducible
$F_4^{(1)}oplus G_2^{(1)}$-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 $eta$-quotients. | | Source: | arXiv, 2206.00163 | | Services: | Forum | Review | PDF | Favorites | |
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