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Growth functions for Artin monoids. (English) Zbl 1183.20061

Summary: In [Proc. Japan Acad., Ser. A 84, No. 10, 179-183 (2008; Zbl 1159.20330)], we showed that the growth function \(P_M(t)\) for an Artin monoid associated with a Coxeter matrix \(M\) of finite type is a rational function of the form \(1/(1-t)N_M(t)\), where \(N_M(t)\) is a polynomial determined by the Coxeter-Dynkin graph for \(M\), and is called the denominator polynomial of type \(M\). We formulated three conjectures on the zeros of the denominator polynomial.
In the present note, we prove that the same denominator formula holds for an arbitrary Artin monoid, and formulate slightly modified conjectures on the zeros of the denominator polynomials of affine types. The new conjectures are verified for types \(\widetilde A_2,\dots,\widetilde A_8\), \(\widetilde C_2,\dots,\widetilde C_8\), \(\widetilde D_4\), \(\widetilde E_7\), \(\widetilde E_8\), \(\widetilde F_4\), \(\widetilde G_2\) among others. In Appendix, we define the elliptic denominator polynomials by formally applying the denominator polynomial formula to the elliptic diagrams for elliptic root systems [Publ. Res. Inst. Math. Sci. 21, 75-179 (1985; Zbl 0573.17012), ibid. 33, No. 2, 301-329 (1997; Zbl 0901.20016)]. Then, the new conjectures are verified also for elliptic denominator polynomials of types \(A_2^{(1,1)},\dots, A_7^{(1,1)}\), \(D_4^{(1,1)}\), \(E_6^{(1,1)}\), \(E_7^{(1,1)}\), \(E_8^{(1,1)}\) and \(G_2^{(1,1)}\).

MSC:

20M05 Free semigroups, generators and relations, word problems
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
17B22 Root systems
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References:

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