TY - JOUR

T1 - Spaces of commuting elements in the classical groups

AU - Kishimoto, Daisuke

AU - Takeda, Masahiro

N1 - Funding Information:
The authors would like to congratulate Akira Kono on the happy occasion of his 70th birthday, and offer this paper as a tribute to him. His work, especially on mapping spaces, have been interesting as well as stimulating for the authors. The first author was supported by JSPS KAKENHI No. 17K05248 .
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/8/6

Y1 - 2021/8/6

N2 - Let G be the classical group, and let Hom(Zm,G) denote the space of commuting m-tuples in G. First, we refine the formula for the Poincaré series of Hom(Zm,G) due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincaré series, and apply it to prove the dependence of the topology of Hom(Zm,G) on the parity of m and the rational hyperbolicity of Hom(Zm,G) for m≥2. Next, we give a minimal generating set of the cohomology of Hom(Zm,G) and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom(Zm,G) with the best possible stable range. Baird proved that the cohomology of Hom(Zm,G) is identified with a certain ring of invariants of the Weyl group of G, and our approach is a direct calculation of this ring of invariants.

AB - Let G be the classical group, and let Hom(Zm,G) denote the space of commuting m-tuples in G. First, we refine the formula for the Poincaré series of Hom(Zm,G) due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincaré series, and apply it to prove the dependence of the topology of Hom(Zm,G) on the parity of m and the rational hyperbolicity of Hom(Zm,G) for m≥2. Next, we give a minimal generating set of the cohomology of Hom(Zm,G) and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom(Zm,G) with the best possible stable range. Baird proved that the cohomology of Hom(Zm,G) is identified with a certain ring of invariants of the Weyl group of G, and our approach is a direct calculation of this ring of invariants.

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U2 - 10.1016/j.aim.2021.107809

DO - 10.1016/j.aim.2021.107809

M3 - Article

AN - SCOPUS:85107135104

SN - 0001-8708

VL - 386

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107809

ER -