Fermionic quasi-particle representations for characters of (G⁽¹⁾)₁ × (G⁽¹⁾)₁ (G⁽¹⁾)₂

We present fermionic quasi-particle sum representations for some of the characters (or branching functions) of (G⁽¹⁾)₁ × (G⁽¹⁾)₁ (G⁽¹⁾)₂ for all simply-laced Lie algebras G. For given G the charactters are written as the partition function of a set of rank G types of massless quasi-particles in certain charge sectors, with nontrivial lower bounds on the one-particle momenta. We discuss the non-uniqueness of the representation for the identity character of the critical Ising model, which arises in both the A₁ and E₈ cases.