On Spaces of Commuting Elements in Lie Groups

Cohen, Fred; Stafa, Mentor; Reiner, V.

doi:10.21236/ada606720

Cited by 5 publications

(16 citation statements)

References 8 publications

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“…Remark There is a different way to package the various

C_{n}^{R}

into a single

C_{2}

‐space, denoted by

{Comm}^{R}

, which has been considered non‐equivariantly in . It is defined via

{prefixComm}^{double-struckR} = (⨆_{n \in double-struckN} C_{n}^{R}) / \sim

with the equivalence relation generated by

false(A_{1}, \dots, A_{n} false) \sim false(A_{1}, \dots, A_{i - 1}, A_{i + 1}, \dots, A_{n} false)

A_{i}

is the identity matrix.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

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Commuting matrices and Atiyah's real K‐theory

Gritschacher

Hausmann

2019

Journal of Topology

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“…Remark There is a different way to package the various

C_{n}^{R}

into a single

C_{2}

‐space, denoted by

{Comm}^{R}

, which has been considered non‐equivariantly in . It is defined via

{prefixComm}^{double-struckR} = (⨆_{n \in double-struckN} C_{n}^{R}) / \sim

with the equivalence relation generated by

false(A_{1}, \dots, A_{n} false) \sim false(A_{1}, \dots, A_{i - 1}, A_{i + 1}, \dots, A_{n} false)

A_{i}

is the identity matrix.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

“…Remark 2.10. There is a different way to package the various C R n into a single C 2 -space, denoted Comm R , that has been considered non-equivariantly in [CS16b]. It is defined via…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Commuting matrices and Atiyah's real K‐theory

Gritschacher

Hausmann

2019

Journal of Topology

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“…Our formula for the Poincaré series of the identity component Hom(Z n , G) 1 builds on work of Baird [4] and Cohen-Reiner-Stafa [15]. In fact, we give a formula for a more refined Hilbert-Poincaré series, which is a tri-graded version of the standard Poincaré series that arises from a certain cohomological description of these spaces due to Baird.…”

Section: Introductionmentioning

confidence: 98%

“…The topology of the spaces Hom(π, G) has been studied extensively in recent years, in particular when π is a free abelian group [2,4,5,16,22,15,14]; in this case Hom(Z n , G) is known as the space of ordered commuting n-tuples in G. The case in which π is a finitely generated nilpotent group was recently analyzed by Bergeron and Silberman [6,7]. These spaces and variations thereon, such as the space of almost commuting elements [9], have been studied in various settings, including work of Witten and Kac-Smilga on supersymmetric Yang-Mills theory [31,32,20].…”

Section: Introductionmentioning

confidence: 99%

Hilbert–Poincaré series for spaces of commuting elements in Lie groups

Ramras¹,

Stafa²

2018

Math. Z.

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In this paper we study homological stability for spaces Hom(Z n , G) of pairwise commuting n-tuples in a Lie group G. We prove that for each n 1, these spaces satisfy rational homological stability as G ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite dimensional analogues of these spaces, Comm(G) and BcomG, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting n-tuples in a fixed group G stabilizes as n increases.Our proofs use the theory of representation stability -in particular, J. Wilson's theory of FI W -modules. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.1.1. Ordered commuting tuples. Let {G r } r 1 denote one of the classical families of compact, connected Lie groups -namely G r = SU(r), U(r), SO(2r + 1), Sp(r), or SO(2r) -or the complexifications thereof. We have standard inclusions G r ֒→ G r+1 (see Section 6).Theorem 1.1. Fix k, n 0 and let {G r } r 1 be one of the classical sequences of Lie groups listed above. Then the standard inclusions G r ֒→ G r+1 induce isomorphismsfor r k, andThese results are proved in Theorems 7.1 and 10.7, and Corollary 11.3 respectively. We prove similar results for G r -equivariant homology in Section 9. This result, and the others discussed below, also apply to the groups Spin(r) (Example 12.2), and to the projectivizations of the above sequences (Example 12.3), although for the projective groups there are no maps inducing the isomorphisms.

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“…, g m are pairwise commutative. The topology of Hom(Z m , G) has been studied intensely in recent years; in particular, its homological and homotopical features have been studied by a number of authors [1,2,4,5,10,11,16,24,25,26]. On the other hand, as in [6], Hom(Z m , G) is identified with the based moduli space of flat G-bundles over an m-torus.…”

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confidence: 99%

Spaces of commuting elements in the classical groups

Kishimoto¹,

Takeda²

2020

Preprint

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Let G be the classical group, and let Hom(Z m , G) denote the space of commuting m-tuples in G. First, we refine the formula for the Poincaré series of Hom(Z m , 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(Z m , G) on the parity of m and the rational hyperbolicity of Hom(Z m , G) for m ≥ 2. Next, we give a minimal generating set of the cohomology of Hom(Z m , G) and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom(Z m , G) with the best possible stable range. Baird proved that the cohomology of Hom(Z m , 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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On Spaces of Commuting Elements in Lie Groups

Cited by 5 publications

References 8 publications

Commuting matrices and Atiyah's real K‐theory

Commuting matrices and Atiyah's real K‐theory

Hilbert–Poincaré series for spaces of commuting elements in Lie groups

Spaces of commuting elements in the classical groups

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