Sets of rigged paths with Virasoro characters

Let {M_r,s^(p,p')}1<r<p-1,1<s<p'-1 be the irreducible Virasoro modules in the (p,p')-minimal series. In our previous paper, we have constructed a monomial basis of ⊕_r=1 ^p-1 M _r,s ^(p,p') in the case 1<p'p<2. By 'monomials' we mean vectors of the form φ(r_L,rL-1)_-nL φ(r1,r0)-n1|r0, s), where φ _-n ^(r',r) :M _r,s ^(p,p') →M _r',s ^(p,p') are the Fourier components of the (2,1)-primary field and |r ₀,s) is the highest weight vector of M(p,p')r0,s. In this article, we introduce for all p<p' with p > 3 and s=1 a subset of such monomials as a conjectural basis of ⊕ _r=1 ^p-1 M _r,1 ^(p,p') . We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.