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Branching rule decomposition of the level1 $E_8^{(1)}$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 KacMoody
counterparts of these algebras and give a subalgebra of type $F_4^{(1)}oplus
G_2^{(1)}$ within a type $E_8^{(1)}$ KacMoody Lie algebra. We will consider
the level1 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 KacPeterson
which uses theta functions and the socalled "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 RogersRamanujan 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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