The moduli of representations of degree 2

Abstract

There are 6 types of 2-dimensional representations in general. For any groups and any monoids, we can construct the moduli of 2-dimensional representations for each type: the moduli of absolutely irreducible representations, representations with Borel mold, representations with semi-simple mold, representations with unipotent mold, representations with unipotent mold over F2, and representations with scalar mold. We can also construct them for any associative algebras.

Virtual Hodge polynomials of the moduli spaces of representations of degree 2 for free monoids

“…Remark 2.4. There exists a scheme Rep 2 (m) ss of finite type over Z by [20], and Rep 2 (m) ss (K) is the associated algebraic variety. By Theorem 2.3, we have Rep n (m) ss (K) ∼ = PGL n (K)/T n × Σn F n (K m ).…”

“…(For * = u, we need to divide Rep 2 (m) u into two parts: the Z[1/2]-part and the F 2 -part. More precisely, see [20]. )…”

“…Let us introduce a general framework for the representation theory over schemes for free monoids (cf. [20,21]). Let Rep n (m) = (M n ) m the product of m-copies of the full n × n matrix ring, which is the representation variety of degree n over Z for the free monoid with m generators.…”

“…The open subscheme Rep n (m) rkn 2 is the moduli space Rep n (m) air of absolutely irreducible representations Rep n (m) air = Rep n (m) rkn 2 . In degree 2 case we can study these moduli spaces in more detail as in [20]. In particular, Rep 2 (m) rk3 is the moduli space Rep 2 (m) B of representations with Borel mold.…”

“…The character variety Ch 2 (m) * is defined as Ch 2 (m) * = Rep 2 (m) * /PGL 2 . In [18], [19], and [20], the first author showed that Ch 2 (m) * is a universal geometric quotient of Rep 2 (m) * by PGL 2 for * = air, B, ss, u, sc. (For * = u, we need to divide Rep 2 (m) u into two parts: the Z[1/2]-part and the F 2 -part.…”