Integrality Properties of Quotients of Wronskians of the Andrews-gordon Series

For positive integers k ≥ 2, we consider the integrality of quotients of Wronskians involving certain normalizations of the Andrews-Gordon q-series ∏ 1≤n 6≡0,±i(mod2k+1) 1 1− qn . This study is motivated by the appearance of these series in conformal field theory as irreducible characters of Virasoro minimal modules. We establish an upper bound on primes p for which such quotients of Wronskians are non-p-integral. Also, for primes p below this bound, we give a criteria for determining the p-integrality of these quotients which depends only on their first few coefficients.