Quantum cluster algebra, braid moves and quantum virtual Grothendieck ring

Abstract

In this paper, we study the quantum virtual Grothendieck ring, denoted by $\frakK_q(\g)$, 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 $\frakK_{q,Q}(\g)$, which corresponds to a simple module over the quiver Hecke algebra $R^\g$, possesses coefficients in $\Z_{\ge 0}[q^{\pm 1/2}]$. This result is particularly interesting because it implies that each truncated Kirillov--Reshetikhin polynomial in $\frakK_{q,Q}(\g)$ and each element in the standard basis $\sfE_q(\g)$ of the entire ring $\frakK_q(\g)$ have coefficients also in $\Z_{\ge 0}[q^{\pm 1/2}]$. 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.