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

Abstract

Abstract We study the family of irreducible modules for quantum affine 𝔰 ⁢ 𝔩 n + 1 {\mathfrak{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 polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to A m {A_{m}} with m ≤ n {m\leq 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 - {\mathscr{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–Hölder 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, 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. Mínguez, Geometric conditions for □ \square -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 {D_{4}} which do not arise from an embedding of A r {A_{r}} with r ≤ 3 {r\leq 3} in D 4 {D_{4}} .