Rank 2 affine MV polytopes

Baumann, Pierre; Dunlap, Thomas; Kamnitzer, Joel; Tingley, Peter

doi:10.1090/S1088-4165-2013-00438-7

Rank 2 affine MV polytopes
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by Pierre Baumann, Thomas Dunlap, Joel Kamnitzer and Peter Tingley

Represent. Theory 17 (2013), 442-468

DOI: https://doi.org/10.1090/S1088-4165-2013-00438-7

Published electronically: August 5, 2013

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Abstract:

We give a realization of the crystal $B(-\infty )$ for $\widehat {\mathrm {sl}}_2$ using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito’s characterization of $B(-\infty )$, in terms of the $*$ involution. The polygons we use have combinatorial properties suggesting they are the $\widehat {\mathrm {sl}}_2$ analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara’s similarity of crystals we also give MV polytopes for $A_2^{(2)}$, the other rank 2 affine Kac-Moody algebra.

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