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A divisibility result on combinatorics of generalized braids

Abstract

For every finite Coxeter group Γ, each positive braids in the corresponding braid group admits a unique decomposition as a finite sequence of elements of Γ, the so-called Garside-normal form. The study of the associated adjacency matrix Adj(Γ) allows to count the number of Garside-normal form of a given length. In this paper we prove that the characteristic polynomial of Adj(B_n) divides the one of Adj(B_(n+1)). The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type A. We observe that this does not hold for type D. The other Coxeter types (I, E, F and H) are also studied.
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hal-01145422 , version 1 (24-04-2015)

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Loic Foissy, Jean Fromentin. A divisibility result on combinatorics of generalized braids. 2015. ⟨hal-01145422⟩
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