Topology of the Representation Varieties With Borel Mold for Unstable Cases

Abstract

In this paper we show that, in the stable case, when m ≥ 2n − 1, the cohomology ring H * (Rep n (m) B ) of the representation variety with Borel mold Rep n (m) B and H * (F n (C m )) ⊗ H * (Flag(C n )) ⊗ Λ(s 1 , . . . , s n−1 ) are isomorphic as algebras. Here the degree of s i is 2m − 3 when 1 ≤ i < n. In the unstable cases, when m ≤ 2n − 2, we also calculate the cohomology group H * (Rep n (m) B ) when n = 3, 4. In the most exotic case, when m = 2, Rep n (2) B is homotopy equivalent to F n (C 2 ) × PGL n (C), where F n (C 2 ) is the configuration space of n distinct points in C 2 . We regard Rep n (2) B as a scheme over Z, and show that the Picard group Pic(Rep n (2) B ) of Rep n (2) B is isomorphic to Z/nZ. We give an explicit generator of the Picard group.2010 Mathematics subject classification: primary 14D22; secondary 20M30, 20C99. Keywords and phrases: moduli of representations, representation variety, Borel mold, free monoid. IntroductionModuli spaces of representations have been investigated and applied by many mathematicians in various subjects. For example, Fricke spaces were constructed as Teichmüller spaces of compact Riemann surfaces of genus g (where g ≥ 2) by using moduli spaces of discrete and faithful representations of the fundamental groups in PSL 2 (R) or SL 2 (R) [1]. The moduli spaces of stable vector bundles on a compact Riemann surface were described as the moduli spaces of irreducible unitary representations of the fundamental group [10,11]. The moduli spaces of θ-semistable representations of quivers were constructed by King [5]. King's construction can be applied to developing the representation theory of wild algebras and to describing moduli spaces of vector bundles on special projective varieties.In this paper, we continue our work from our paper [9] to investigate the topology of representation varieties with Borel mold more precisely. The objects with which we deal here are not irreducible representations, but representations with Borel mold. By a global representation theory we mean a theory of representations parametrized by schemes or topological spaces. Recall that a mold is a subsheaf of O X -subalgebras of the full matrix ring M n (O X ) that is a subbundle of M n (O X ) on ringed spaces (X, O X ). Several moduli spaces of representations have been constructed for given types of molds. For example, in [7] we treat the moduli space of absolutely irreducible representations, and in [6] we treat the moduli space of representations with Borel mold. We propose to construct the moduli space with any mold of degree two for general groups and monoids in a future paper.Recall that a representation with Borel mold for a group or a monoid is a representation that can be normalized to a representation in upper triangular matrices and whose image of the group or the monoid generates the algebra of upper triangular matrices. The moduli space of representations with Borel mold is important, since representations with Borel mold are typical examples of indecomposable representation...

Quasi-flat representations of uniform groups and quantum groups

“…Matrix models provide another possible connection to the literature via the study of character or representation varieties for discrete groups [22,23,28,29,31,33]. These are spaces (typically algebraic varieties) parametrizing the linear representations of Γ in much the same way that our space X above does.…”