We exhibit quantum cluster algebra structures on quantum Grassmannians Kq[Gr(2,n)] and their quantum Schubert cells, as well as on Kq[Gr(3,6)], Kq[Gr(3,7)] and Kq[Gr(3,8)]. These cases are precisely those where the quantum cluster algebra is of finite type and the structures we describe quantize those found by Scott for the classical situation.
Notes on the B+B+t theorem
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