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

We study the family of irreducible modules for quantum affine𝔰𝔩n+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 toAm{A_{m}}withmn{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 category𝒞-{\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 toq-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 typeD4{D_{4}}which do not arise from an embedding ofAr{A_{r}}withr3{r\leq 3}inD4{D_{4}}.

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Journal für die reine und angewandte Mathematik (Crelles Journal)
Medium: X
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National Science Foundation
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