Let m,m′∈N and j,j′∈Z. For m∈N and j∈Z, let CHN=3(+)[K(m),j] denote the C((q21))-linear span of the N=3 characters chH(Λ[K(m),m2])(+)(τ,z) with m2∈j+2Z and 0≤m2≤m. Product-closure conjecture. The following inclusion will hold for all m,m′∈N and j,j′∈Z:
CHN=3(+)[K(m),j]⋅CHN=3(+)[K(m′),j′]⊂CHN=3(+)[K(m+m′),j+j′].
Equivalently, for m2,m2′∈Z satisfying 0≤m2≤m and 0≤m2′≤m′, there should exist functions bm2”(m,m2),(m′,m2′)∈C((q21)) such that
chH(Λ[K(m),m2])(+)(τ,z)chH(Λ[K(m′),m2′])(+)(τ,z)=m2”∈m2+m2′+2Z0≤m2”≤m+m′∑bm2”(m,m2),(m′,m2′)(τ)chH(Λ[K(m+m′),m2”])(+)(τ,z).
This predicts that products of character spaces are closed under multiplication, with the parameters m and j adding. The preceding proposition establishes several special product inclusions, while the general product formula remains conjectural.
