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Register a new userMotive of the representation varieties of torus knots for low rank affine groups
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We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C].
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We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interplay between these two results in sight of Katz theorem that relates the point count polynomial with the E-polynomial. In particular, we shall show that several point count polynomials exist for these representation varieties, depending on the arithmetic between m,n and the characteristic of the field, whereas only one of them agrees with the actual E-polynomial.
In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the computation of the motive to a combinatorial problem.
In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent and arbitrary elements of mathfrak{gl}_n(C) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases the Hessenberg variety has no odd dimensional cohomology.
The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group.
We extend the Altmann-Hausen presentation of normal affine algebraic C-varieties endowed with effective torus actions to the real setting. In particular, we focus on actions of quasi-split real tori, in which case we obtain a simpler presentation.
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