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Shortest path poset of finite Coxeter groups

Abstract : We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$.
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Saúl A. Blanco. Shortest path poset of finite Coxeter groups. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.189-200, ⟨10.46298/dmtcs.2721⟩. ⟨hal-01185413⟩

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