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The growth series of the \(n\)-extended affine Weyl group of type \(A_1\). (English) Zbl 1084.20028

Summary: \(n\)-extended affine Weyl groups are Weyl groups associated to \(n\)-extended affine root systems introduced by K. Saito [Publ. Res. Inst. Math. Sci. 21, 75-179 (1985; Zbl 0573.17012)]. We calculate the growth series of the \(n\)-extended affine Weyl group of type \(A_1\) with a generator system of an \(n\)-toroidal sense.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
17B20 Simple, semisimple, reductive (super)algebras

Citations:

Zbl 0573.17012
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References:

[1] K. Saito, Extended affine root systems. I. Coxeter transformations, Publ. Res. Inst. Math. Sci. 21 (1985), no. 1, 75-179. · Zbl 0573.17012 · doi:10.2977/prims/1195179841
[2] K. Saito and T. Takebayashi, Extended affine root systems. III. Elliptic Weyl groups, Publ. Res. Inst. Math. Sci. 33 (1997), no. 2, 301-329. · Zbl 0901.20016 · doi:10.2977/prims/1195145453
[3] R. P. Stanley, Enumerative combinatorics. Vol. I , The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Adv. Books Software, Monterey, CA, 1986.
[4] M. Wakimoto, Poincaré series of the Weyl group of elliptic Lie algebras \(A_1^{(1,1)}\) and \(A_1^{(1,1)*}\), q-alg/9705025.
[5] T. Takebayashi, Poincaré series of the Weyl groups of the elliptic root systems \(A_1^{(1,1)},\;A_1^{(1,1)*}\) and \(A_2^{(1,1)}\), J. Algebraic Combin. 17 (2003), no. 3, 211-223. · Zbl 1047.20030 · doi:10.1023/A:1025081404009
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