Abstract
Kronecker coefficients encode the tensor products of complex irreducible representations of symmetric groups. Their stability properties have been considered recently by several authors (Vallejo, Pak and Panova, Stembridge). We describe a geometric method, based on Schur–Weyl duality, that allows to produce huge series of instances of this phenomenon. Moreover, the method gives access to lots of extra information. Most notably, we can often compute the stable Kronecker coefficients, sometimes as numbers of points in very explicit polytopes. We can also describe explicitly the moment polytope in the neighbourhood of our stable triples. Finally, we explain an observation of Stembridge on the behaviour of certain rectangular Kronecker coefficients, by relating it to the affine Dynkin diagram of type \(E_6\).
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Acknowledgments
This paper was begun in Berkeley during the semester on Algorithms and Complexity in Algebraic Geometry organized at the Simons Institute for Computing, and completed in Montréal at the Centre de Recherches Mathématiques (Université de Montréal) and the CIRGET (UQAM). The author warmly thanks these institutions for their generous hospitality.
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Manivel, L. On the asymptotics of Kronecker coefficients. J Algebr Comb 42, 999–1025 (2015). https://doi.org/10.1007/s10801-015-0614-1
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DOI: https://doi.org/10.1007/s10801-015-0614-1