Quantum cluster algebra, braid moves and quantum virtual Grothendieck ring

In this paper, we study the quantum virtual Grothendieck ring, denoted by q (), which was introduced in 39, and further investigated in 26, 25. 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 has 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 ₐ, ₐ (), which corresponds to a simple module over the quiver Hecke algebra R^, possesses coefficients in ₀q^ 1/2. This result is particularly interesting because it implies that each truncated Kirillov--Reshetikhin polynomial in ₐ, ₐ () and each element in the standard basis q () of the entire ring q () have coefficients also in ₀q^ 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.