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تسجيل مستخدم جديدFibers of maps to totally nonnegative spaces
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This paper undertakes a study of the structure of the fibers of a family of maps $f_{(i_1,dots ,i_d)}$ arising from representation theory, motivated both by connections to Lusztigs theory of canonical bases and also by the fact that these fibers encode the nonnegative real relations amongst exponentiated Chevalley generators. In particular, we prove that the fibers of these maps $f_{(i_1,dots ,i_d)}$ (restricted to the standard simplex $Delta_{d-1}$ in a way that still captures the full structure) admit cell decompositions induced by the decomposition of $Delta_{d-1}$ into open simplices of various dimensions. We also prove that these cell decompositions have the same face posets as interior dual block complexes of subword complexes and that these interior dual block complexes are contractible. We conjecture that each such fiber is a regular CW complex homeomorphic to the interior dual block complex of a subword complex. We show how this conjecture would yield as a corollary a new proof of the Fomin-Shapiro Conjecture by way of general topological results regarding approximating maps by homeomorphisms.
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A map $X$ on a surface is called vertex-transitive if the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In general, s
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If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map $X$ on the
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A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Cl
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