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In [FFR] Feingold, Frenkel and Ries gave a spinor construction of the vertexoperator para-algebra (abelian intertwining algebra) V = V^0 \oplus V^1 \oplusV^2 \oplus V^3, whose summands are four level-1 irreducible representations ofthe affine Kac-Moody algebra D_4^(1). The triality group S_3 = < \sigma,\tau |\sigma^3 = 1 = \tau^2, \tau\sigma\tau = \sigma^{-1} > in Aut(V) wasconstructed, preserving V^0 and permuting the V^i, for i=1,2,3. V is(1/2)Z-graded where V^i_n denotes the n-graded subspace of V^i. Vertexoperators Y(v,z) for v in V^0_1 represent D_4^(1) on V, while those for which\sigma(v) = v represent G_2^(1).  We investigate branching rules, how V decomposes into a direct sum ofirreducible G_2^(1) representations. We use a two-step process, firstdecomposing with respect to the intermediate subalgebra B_3^(1), represented byY(v,z) for \tau(v) = v. There are three vertex operators, Y(\omega_{D_4},z),Y(\omega_{B_3},z), and Y(\omega_{G_2},z) each representing the Virasoro algebragiven by the Sugawara constructions from the three algebras. TheGoddard-Kent-Olive coset construction [GKO] gives two mutually commuting cosetVirasoro representations, provided by the vertex operatorsY(\omega_{D_4}-\omega_{B_3},z) and Y(\omega_{B_3}-\omega_{G_2},z), with centralcharges 1/2 and 7/10, respectively. The first one commutes with B_3^(1), andthe second one commutes with G_2^(1). This gives the space of highest weightvectors for G_2^(1) in V as tensor products of irreducible Virasoro modulesL(1/2,h^{1/2}) \otimes L(7/10,h^{7/10}). This dissertation explicitlyconstructs these coset Virasoro operators, and uses them to describe thedecomposition of V with respect to G_2^(1). This work also provides spinorconstructions of the 7/10 Virasoro modules, and of the two level-1representations of G_2^(1) inside V.

Decomposition of Level-1 Representations of D_4^(1) With Respect to its Subalgebra G_2^(1) in the Spinor Construction