Piecewise-linear and birational toggling
نویسندگان
چکیده
We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset P as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley’s transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When P = [a] × [b], a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order a + b. We prove some homomesy results, showing that for certain functions f , the average of f over each rowmotion/promotion orbit is independent of the orbit chosen. Résumé. Nous définissons et étudions certains analogues linéaires-par-morceaux et birationnels d’involutions toggles, rowmotion et promotion sur les idéaux d’un poset P , comme étudié par Striker et Williams. La rowmotion linéairepar-morceaux est liée à la fonction transfert de Stanley pour les polytopes d’ordre; la promotion linéaire-par-morceaux se rapporte à la promotion de Schützenberger pour les tableaux semistandards de Young. Lorsque P = [a]× [b], une propriété de symétrie réciproque récemment prouvée par Grinberg et Roby implique que la rowmotion birationnelle (et par conséquent la rowmotion linéaire-par-morceaux) est de l’ordre a + b. Nous démontrons quelques résultats d’homomesie, montrant que pour certaines fonctions f , la moyenne de f sur chaque orbite de rowmotion/promotion est indépendante de l’orbite choisie.
منابع مشابه
The order of birational rowmotion
Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called rowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe h...
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