The moduli of representations with Borel mold

Abstract

The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.

The moduli of representations of degree 2

“…Theorem 2.10 ( [12]). There exists a fine moduli scheme Ch n (Γ) B separated over Z associated to the sheafification EqB n (Γ) of the following functor with respect to Zariski topology:…”

“…In [12] we have introduced the notion of mold. A mold is, so to say, a subalgebra of the full matrix ring.…”

“…For example, we have constructed the moduli of equivalence classes of absolutely irreducible representations denoted by Ch n (Γ) air in [10], where ρ : Γ → M n (Γ(X, O X )) is absolutely irreducible if O X [ρ(Γ)] coincides with the full matrix ring M n (O X ). We also have constructed the moduli of equivalence classes of representations with Borel mold denoted by Ch n (Γ) B in [12], where ρ : Γ → M n (Γ(X, O X )) is a representation with Borel mold if for each x ∈ X there exists P ∈ GL n (O X (U)) on a neighbourhood U of x such that P · O U [ρ(Γ)] · P −1 coincides with the subsheaf of O U -algebras of M n (O U ) consisting of upper triangular matrices. The author calls the plan to construct the moduli of equivalence classes of representations for any suitable molds "mold program".…”

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

“…Although they are of great importance in the representation theory over schemes, it is difficult to analyze them. To overcome this difficulty, the first author introduced representations with Borel mold and studied the representation variety and the character variety of representations with Borel mold in [19]. Furthermore, we studied the topology of these moduli spaces of representations with Borel mold for free monoids over the field C of complex numbers in [21].…”

“…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.…”