Rational homotopy type of the moduli of representations with Borel mold (Q2890541)

The authors investigate representations of free algebras or free monoids. Let \(\text{Rep}_n(m)_B\) be the moduli space of representations with Borel mold of degree \(m\) for the free monoid with \(n\) generators, and let \(B_n(m)_B\) be a subscheme of \(\text{Rep}_n(m)_B\) which consists of the representations with Borel mold in upper triangular matrices. The associated character variety of \(\text{Rep}_n(m)_B\) is denoted \(\text{Ch}_n(m)_B\).NEWLINENEWLINE The first main theorem in this note is: If \(K\) is one of the three complex algebraic varieties considered above then \(K\) is rationally homotopy equivalent to the product of the configuration space of \(n\) distinct points in \(\mathbb{C}^n\) and the product of \((n-1)\) copies of the projective space \(\mathbb{C} P^{m-2}\). The second main theorem describes the Sullivan minimal model of \(K\) with mixed Hodge structure as formerly studied by Morgan.