Topology of the Moduli of Representations with Borel Mold

A representation for a group or a monoid is called a representation with Borel mold if it can be normalized to a representation in upper triangular matrices whose image of the group or monoid generates the algebra of upper triangular matrices. In [Na2] the moduli of representations with Borel mold has been constructed for each group or monoid. The moduli of representations with Borel mold has simpler structure than the moduli of absolutely irreducible representations constructed in [Na1]. In the present paper, for the free monoid case we describe the moduli of representations with Borel mold explicitly, and calculate its cohomology ring. The moduli of representations with Borel mold has a fibre bundle structure over the configuration space of the affine space, and hence its cohomology ring can be calculated. We also calculate the virtual Hodge polynomial of the moduli of representations with Borel mold, which will be used for calculating the virtual Poincaré polynomial of the moduli of absolutely irreducible representations of degree 2 for the free monoid case in [Na3]. By calculating the cohomology ring of the moduli, we can consider characteristic classes for representations with Borel mold on a scheme. The construction of characteristic classes and its application will be presented in other papers. By global representation theory we understand theory of representations on (arbitrary) schemes. The global representation theory is geometric rather than the local representation theory, that is, the representation theory over fields or local rings. For example, each representation of degree n with Borel mold for a group (or a monoid) Γ on a scheme X has a unique Γ-invariant complete flag of On X (see [Na2]). The Γ-invariant complete flag is not always trivial on X, although if X is the spectrum of a field or a local ring, then the flag is trivial. Non-triviality of the Γ-invariant flag is an interesting feature of the theory of representations over schemes.