2024/02/13 by Lee, Kyu-Hwan, Oh, Se-jin
#13F60 #17B10 #17B37 #17B67 #18N25 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2402.08140
In this paper, we study the quantum virtual Grothendieck ring, denoted by \frakKq(\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 \frakKq,Q(\g), which corresponds to a simple module over the quiver Hecke algebra R^\g, possesses coefficients in \Z≥ 0[q± 1/2]. This result is particularly interesting because it implies that each truncated Kirillov--Reshetikhin polynomial in \frakKq,Q(\g) and each element in the standard basis \sfEq(\g) of the entire ring \frakKq(\g) have coefficients also in \Z≥ 0[q± 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.