In this paper, we study the quantum virtual Grothendieck ring, denoted by \(\mathfrak {K}_q(\mathfrak {g})\) , which was introduced in [39], and further investigated in [25, 26]. Our approach involves examining this ring from two perspectives: first, by considering its connection to quantum cluster algebras of non-skew-symmetric types; and second, by exploring its relevance to categorification theory. We specifically focus on (i) the homomorphisms that arise from braid moves, particularly 4-moves and 6-moves, in the braid group; and (ii) the quantum Laurent positivity phenomena, which have not yet been proven for non-skew-symmetric types. As applications of our results, we derive the substitution formulas for non-skew-symmetric types discussed in [11] for skew-symmetric types, and demonstrate that any truncated element in a heart subring, denoted by \(\mathfrak {K}_{q,Q}(\mathfrak {g})\) , which corresponds to a simple module over the quiver Hecke algebra \(R^\mathfrak {g}\) , possesses coefficients in \(\mathbb {Z}_{\geqslant 0}[q^{\pm 1/2}]\) . This result is particularly interesting because it implies that each truncated Kirillov–Reshetikhin polynomial in \(\mathfrak {K}_{q,Q}(\mathfrak {g})\) and each element in the standard basis \(\textsf{E}_q(\mathfrak {g})\) of the entire ring \(\mathfrak {K}_q(\mathfrak {g})\) have coefficients also in \(\mathbb {Z}_{\geqslant 0}[q^{\pm 1/2}]\) . Since (truncated) Kirillov–Reshetikhin polynomials can be obtained using a quantum cluster algebra algorithm and appear as quantum cluster variables, they provide compelling evidence in support of the quantum Laurent positivity conjecture in non-skew-symmetric types.
