A note on nilpotent representations

Bergeron, Maxime; Silberman, Lior

doi:10.1515/jgth-2015-0027

Cited by 11 publications

(19 citation statements)

References 16 publications

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“…It follows that the third vertical map is also a homotopy equivalence. The first vertical map and the second vertical maps are G-equivariant by results of Bergeron and Silberman [7]. Therefore, obtain a commutative diagram S n,2 (G)/G Hom(Z n , G) 1 /G Hom(Z n , G) 1 /G = T n /W S n,q (G)/G Hom(F n /Γ q n , G) 1 /G Hom(F n /Γ q n , G) 1 /G.…”

mentioning

confidence: 63%

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Poincaré series of character varieties for nilpotent groups

Stafa

2018

Journal of Group Theory

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For any compact and connected Lie group G and any free abelian or free nilpotent group Γ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety) Rep(Γ, G) 1 , with coefficients in a field F with char(F) either 0 or relatively prime to the order of the Weyl group W . We give explicit formulas for the Poincaré series. In addition we study G-equivariant stable decompositions of subspaces X(q, G) of the free monoid J(G) generated by the Lie group G, obtained from finitely generated free nilpotent group representations.

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mentioning

confidence: 63%

“…Bergeron and Silberman [7] show that if Γ is a finitely generated nilpotent group, then Rep(Γ, G) 1 has the same homology as Rep(Γ/Γ q , G) 1 with coefficients in a field with characteristic not dividing |W |. Therefore, Theorem 1.4 applies also to all such Γ, in particular to free nilpotent groups F n /Γ q .…”

mentioning

confidence: 96%

Poincaré series of character varieties for nilpotent groups

Stafa

2018

Journal of Group Theory

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“…The corresponding result for Rep(Z n , G) was established by Florentino-Lawton [22] (note that when G is reductive, Rep(Z n , G) should be interpreted as a GIT quotient), and generalizations to nilpotent groups discrete groups were obtained by Bergeron [8]. Moreover, Bergeron and Silberman showed in [9] that when G is compact and Γ is nilpotent, the connected component of the identity in Hom(Γ, G) contains only abelian representations, and hence is the same as Hom(H 1 (Γ), G) 1 . These results allow us to extend most of the stability statements in the paper (Section 11).…”

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confidence: 88%

“…By Lemmas 7.6 and 7.7 together with Theorems 4.11 and 4.20, it suffices to show that the FI#-modules H k (P(T )) W and H k (G/T × P(T )) W are generated in stage k. Proposition 5.3 tells us that H k (P(T )) is finitely generated in stage k, and the same holds for H k (P(T )) W by Lemma 7.8. Next, we have a decomposition of FI#-modules (9) H…”

Section: 3mentioning

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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“…Also, for a fixed m > 2 one could ask for stability conditions on q → B(q, SU (m)). For the reduced version, B(q, SU (m)) 1 → B(q + 1, SU(m)) 1 is a homeomorphism for every q ≥ 2 by a result of M. Bergeron and L. Silberman [10]. But this is no longer the case for the full non-(−) 1 version.…”

Section: Homology Of B(q Su (2))mentioning

confidence: 99%

Nilpotent n-tuples in SU(2)

Camarena

Villarreal

2020

Proceedings of the Edinburgh Mathematical Society

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We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$ . The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$ . We give explicit calculations when $\Gamma$ is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration $B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq 2$ , the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.

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A note on nilpotent representations

Cited by 11 publications

References 16 publications

Poincaré series of character varieties for nilpotent groups

Poincaré series of character varieties for nilpotent groups

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

Nilpotent n-tuples in SU(2)

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