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

Abstract

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.

Mixed Hodge structures on character varieties of nilpotent groups

“…Remark 9. The characteristic exponents referenced in Theorem 8 coincide with the ones for a maximal compact K ⊂ G. Therefore, these are well-known for all simple G (see [RS,Table 1] or the table in [KT,Page 7]).…”

“…The formula for M 0 Γ G then follows from the one in [FS]. To get the formula for R 0 Γ G we observe, as in [RS,Section 5], that the graded cohomology ring of R Z r K is a regrading of the cohomology ring of the torus T r . Using representation theory, analogous to what is done in [KT], we determine the regrading explicitly to obtain Formula (1.1).…”

Commuting matrices and Atiyah's real K‐theory

“…Ramras and Stafa show in [31,Theorem 7.7] that the rational homology groups of the spaces C n (U (m)), Rep n (U (m)) as well as C n,1 (O(m)), Rep n,1 (O(m)) -for fixed m and varying n -are representation stable in the sense of [12], with stability range independent of m. The homology groups also satisfy homological stability in m, by [31, Theorem 1.2]; hence, it follows that the rational homology groups of the spaces C n (U ), Rep n (U ), C n (O) and Rep n (O) are also representation stable in n. One might wonder whether the (integral) homotopy groups π k (−) of these spaces are also representation stable. The answer is 'yes' in the unitary case, whereas in the orthogonal case, it turns out to depend on k.…”

“…, g n ) ∈ G ×n | g i g j = g j g i for all i, j}, viewed as a subspace of G ×n , or, in other words, the space of group homomorphisms Z n → G. Closely related is the representation space Rep n (G), which is defined as the quotient of C n (G) by the action of G by conjugation. These spaces are of classical interest because of their relevance in mathematical physics [24,39], and also have interesting homotopy-theoretic properties: For example, they admit stable splittings [2,8], they give rise to the spectrum of commutative K-theory [4][5][6], and they satisfy rational homological stability when G runs through any of the sequences of classical Lie groups [31]. We refer to [14] for a survey.…”

Torsion in the space of commuting elements in a Lie group

“…and so it is often called the space of commuting m-tuples in G. The topology of Hom(Z m , G) has been intensely studied in recent years. See for example [1,2,3,4,5,11,13,17,20,21]. In this paper, we study the homology of Hom(Z m , G).…”