Quantum cluster algebra, braid moves and quantum virtual Grothendieck ring
Kyu-Hwan Lee, Se-jin Oh
TL;DR
The work addresses the quantum virtual Grothendieck ring $K_q(g)$ by embedding it into a quantum cluster algebra framework and by categorifying it through quiver Hecke algebras. It analyzes braid-move mutations (including 4- and 6-moves) and forward shifts to establish degree-compatibility, and it derives substitution formulas that preserve the canonical basis across seeds. The main contributions include proofs of quantum positivity for fundamental polynomials, standard basis elements, and truncated Kirillov–Reshetikhin polynomials in non-skew-symmetric types, providing strong evidence for quantum Laurent positivity in this setting. Together with the categorification perspective, these results illuminate the structure of $K_q(g)$ and its connections to cluster algebras and representation theory, with implications for monoidal categorification and positivity phenomena.
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.