Abstract
One way to study the Kronecker coefficients is to focus on the Kronecker cone, which is generated by the triples of partitions corresponding to non-zero Kronecker coefficients. In this article we are interested in producing particular faces of this cone, formed of stable triples (a notion defined by J. Stembridge in 2014), using some geometric notions—principally those of dominant and well-covering pairs—and results of N. Ressayre. This extends a result obtained independently by L. Manivel and E. Vallejo in 2014 or 2015, expressed in terms of additive matrices. To illustrate the fact that it allows to produce quite a few new faces of the Kronecker cone, we give at the end of the article details about what our results yield for “small dimensions”.
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Notes
This assumption is in fact not a restriction on the faces that we produce at the end.
Note that P and \({\hat{P}}\) are omitted from this notation, even though \({\mathcal {F}}(C)\) of course also depends on them.
He even proved it in a much more general setting than simply \(\mathrm {PKron}\).
See Remark 3.7 for details on that.
The ordering of the basis of \(V_1\otimes V_2\) gives in particular an explicit bijection between \(\llbracket 1,n_1\rrbracket \times \llbracket 1,n_2\rrbracket \) and \(\llbracket 1,n_1n_2\rrbracket \), which we will regularly use to identify the two in what follows.
In what follows, for any \({\hat{u}}\in {\hat{W}}\), \({\hat{u}}^\vee \) is defined as \({\hat{w}}_0{\hat{u}}\), where \({\hat{w}}_0\) is the longest element of the Weyl group \({\hat{W}}\).
The set of one-parameter subgroups of T verifying condition (1) is an open convex polyhedral cone and, among those subgroups, the not \({\hat{G}}\)-regular ones are elements of some hyperplanes. Thus the set of dominant \({\hat{G}}\)-regular one-parameter subgroups of T verifying condition (1) is not empty.
For the interested reader, they can be found online at http://math.univ-lyon1.fr/~pelletier/recherche/matrices_additives_3x3.pdf, along with the number of well-covering and dominant pairs that each provides.
References
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Acknowledgements
This article was mainly written while working at the Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France. I would like to thank Nicolas Ressayre for his advice and for invaluable discussions during the preparation of this article. I also acknowledge support from the French ANR (ANR project ANR-15-CE40-0012).
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Pelletier, M. Production of faces of the Kronecker cone containing stable triples. Geom Dedicata 214, 739–765 (2021). https://doi.org/10.1007/s10711-021-00634-x
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DOI: https://doi.org/10.1007/s10711-021-00634-x