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Abstract The pentagram map on polygons in the projective plane was introduced by R. Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. In the present paper we introduce and prove integrability of long‐diagonal pentagram maps on polygons in , by now the most universal pentagram‐type map encompassing all known integrable cases. We also establish an equivalence of long‐diagonal and bi‐diagonal maps and present a simple self‐contained construction of the Lax form for both. Finally, we prove that the continuous limit of all these maps is equivalent to the ‐KdV equation, generalizing the Boussinesq equation for .
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Abstract For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ . Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ . Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.more » « less
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The pentagram map takes a planar polygon $$P$$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $$P$$ . This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $$P$$ is a Poncelet polygon, then the image of $$P$$ under the pentagram map is projectively equivalent to $$P$$ . In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.more » « less
