Physical Combinatorics pp 49-103 | Cite as

# On the Combinatorics of Forrester-Baxter Models

## Abstract

We provide further boson-fermion *q*-polynomial identities for the “finitized” Virasoro characters
χ _{ r.s } ^{ p.p′ } of the Forrester—Baxter minimal models *M* (*p*, *p*′) for certain values of *r* and *s*. The construction is based on a detailed analysis of the combinatorics of the set
\(\mathcal{P}_{{a.b.c}}^{{p.p'}}(L)\) of *q*-weighted, length-*L* Forrester-Baxter paths, whose generating function
\(X_{a.b.c}^{p.p'} \) (*L*) provides a finitization of χ _{ r.s } ^{ p.p′ } . In this paper, we restrict our attention to the case where the startpoint *a* and endpoint *b* of each path both belong to the set of *Takahashi lengths*. In the limit *L* → ∞, these polynomial identities reduce to *q*-series identities for the corresponding characters.

We obtain two closely related fermionic polynomial forms for each (finitized) character. The first of these forms uses the classical definition of the Gaussian polynomials and includes a term that is a (finitized) character of a certain \(M(\hat{p},\hat{p}')\), where \(\hat{p}' < p'\). We provide a combinatorial interpretation for this form using the concept of *particles*. The second form, which was first obtained using different methods by the Stony Brook group, requires a modified definition of the Gaussian polynomials, and its combinatorial interpretation requires not only the concept of particles, but also the additional concept of *particle annihilation*.

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