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Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux

Joseph PappeStephan PfannererAnne SchillingMary Claire Simone
Dec 2022
摘要
We construct an injection from the set of $r$-fans of Dyck paths (resp.vacillation tableaux) of length $n$ into the set of chord diagrams on $[n]$that intertwines promotion and rotation. This is done in two different ways,namely as fillings of promotion--evacuation diagrams and in terms of Fomingrowth diagrams. Our analysis uses the fact that $r$-fans of Dyck paths andvacillating tableaux can be viewed as highest weight elements of weight zero incrystals of type $B_r$ and $C_r$, respectively, which in turn can be analyzedusing virtual crystals. On the level of Fomin growth diagrams, thevirtualization process corresponds to Krattenthaler's blow up construction. Oneof the motivations for finding rotation invariant diagrammatic bases such aschord diagrams is the cyclic sieving phenomenon. Indeed, we give a cyclicsieving phenomenon on $r$-fans of Dyck paths and vacillating tableaux using thepromotion action.
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