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On truncated Weyl modules. (English) Zbl 07052405
Summary: We study structural properties of truncated Weyl modules. A truncated Weyl module \(W_N(\lambda)\) is a local Weyl module for \(\mathfrak{g}[t]_N=\mathfrak{g}\otimes\frac{\mathbb{C}[t]}{t^N\mathbb{C}[t]}\), where \(\mathfrak{g}\) is a finite-dimensional simple Lie algebra. It has been conjectured that, if \(N\) is sufficiently small with respect to \(\lambda\), the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when \(\lambda\) is a multiple of certain fundamental weights, including all minuscule ones for simply laced \(\mathfrak{g}\). We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that \(\mathfrak{g}=\mathfrak{sl}_2\) related to Demazure flags and chains of inclusions of truncated Weyl modules.
MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B70 Graded Lie (super)algebras
05E10 Combinatorial aspects of representation theory
06A07 Combinatorics of partially ordered sets
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