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Register a new userOn the action of the toggle group of the Dynkin diagram of type A
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In this article, we consider involutions, called togglings, on the set of independent sets of the Dynkin diagram of type A, or a path graph. We are interested in the action of the subgroup of the symmetric group of the set of independent sets generated by togglings. We show that the subgroup coincides with the symmetric group.
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Baker and Wang define the so-called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but it is independent of the base vertex if and only if the ribbon structure is planar, moreover, in this case the Bernardi action is compatible with planar duality. Earlier, Chan, Church and Grochow and Chan, Glass, Macauley, Perkinson, Werner and Yang proved analogous results about the rotor-routing action. Baker and Wang moreover showed that the Bernardi and rotor-routing actions coincide for plane graphs. We clarify this still confounding picture by giving a canonical definition for the planar Bernardi/rotor-routing action, and also a canonical isomorphism between sandpile groups of planar dual graphs. Our canonical definition implies the compatibility with planar duality via an extremely short argument. We also show hidden symmetries of the problem by proving our results in the slightly more general setting of balanced plane digraphs. Any balanced plane digraph gives rise to a trinity, i.e., a triangulation of the sphere with a three-coloring of the $0$-simplices. Our most important tool is a group associated to trinities, introduced by Cavenagh and Wanless, and a result of a subset of the authors characterizing the Bernardi bijection in terms of a dissection of a root polytope.
Consider equipping an alphabet $mathcal{A}$ with a group action that partitions the set of words into equivalence classes which we call patterns. We answer standard questions for the Penneys game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.
For a map $mathcal M$ cellularly embedded on a connected and closed orientable surface, the bases of its Lagrangian (also known as delta-) matroid $Delta(mathcal M)$ correspond to the bases of a Lagrangian subspace $L$ of the standard orthogonal space $mathbb{Q}^Eoplusmathbb{Q}^{E^*}$, where $E$ and $E^*$ are the edge-sets of $mathcal M$ and its dual map. The Lagrangian subspace $L$ is said to be a representation of both $mathcal M$ and $Delta(mathcal M)$. Furthermore, the bases of $Delta(mathcal M)$, when understood as vertices of the hypercube $[-1,1]^n$, induce a polytope $mathbf P(Delta(mathcal M))$ with edges parallel to the root system of type $BC_n$. In this paper we study the action of the Coxeter group $BC_n$ on $mathcal M$, $L$, $Delta(mathcal M)$ and $mathbf P(Delta(mathcal M))$. We also comment on the action of $BC_n$ on $mathcal M$ when $mathcal M$ is understood a dessin denfant.
We give upper bounds on the order of the automorphism group of a simple graph
We show that the mapping class group acts properly on the space of maximal representations of the fundamental group of a closed Riemann surface into G when G = Sp(2n,R), SU(n,n), SO*(2n) or Spin(2,n).
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