Higher order Kirillov--Reshetikhin modules for q (A n (1)), imaginary modules and monoidal categorification

We study the family of irreducible modules for quantum affine sl(n+1) whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polyno-mials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to A(m) with m <= n. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category C -. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov-Reshetikhin module with its dual always contains an imaginary module in its Jordan-H & ouml;lder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc, A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 5, 1113-1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. M & iacute;nguez, Geometric conditions for -irreducibility of certain representations of the general linear group over a non-archimedean local field, Adv. Math. 339 (2018), 113-190]. Finally, we use our methods to give a family of imaginary modules in type D-4 which do not arise from an embedding of A(r) with r <= 3 in D-4.