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Sean Lawton

Publications and source records attributed to Sean Lawton.

At least 37 records · Page 2Linked to original sources
arXiv2016

Character Varieties of Free Groups are Gorenstein but not always Factorial

Fix a rank g free group F and a connected reductive complex algebraic group G. Let X(F,G) be the G-character variety of F. When the derived subgroup DG in G is simply connected we show that X(F,G) is factorial (which implies it is Gorenstein), and provide examples to show that when DG is not simply connected X(F,G) need not even be locally factorial. Despite the general failure of factoriality of these moduli spaces, using different methods, we show that X(F,G) is always Gorenstein.

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arXiv2015

E-polynomial of the SL(3,C)-character variety of free groups

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3, C). Expanding upon existing techniques, we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give new proofs of the SL(2,C), PGL(2,C), and PGL(3,C) cases.

math.AG↗
arXiv2015

Fundamental Group of Moduli Spaces of Representations

Let S be a surface of genus g with n points removed, G a connected Lie group, and X(G) the moduli space of representations of the fundamental group of S into G. We compute the fundamental group of X(G) when n>0 and G is a real or complex reductive algebraic group, or a compact Lie group; and when n=0 and G=GL(m,C), SL(m,C), U(m), or SU(m).

math.AG↗
arXiv2014

Homotopy type of free group character varieties

Let G be a real reductive algebraic group with maximal compact subgroup K, and let F be a rank r free group. Here, we summarize the construction of a natural strong deformation retraction from the space of closed orbits in Hom(F,G)/G to the orbit space Hom(F,K)/K. In particular, these spaces have the same homotopy type.

math.AT↗
arXiv2014

SL(3,C)-Character Varieties and RP2-Structures on a Trinion

Denote the free group on two letters by F2 and the SL(3,C)-representation variety of F2 by R = Hom(F2, SL(3, C)). There is a SL(3,C)-action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. We determine explicit minimal generators and defining relations for the subring of invariants and show X is a hyper-surface in C9. Our choice of generators exhibit Out(F2) symmetries which allow for a succinct expression of the defining relations. We then show C[X] is a Poisson algebra with respect to a presentation of F2 imposed by a punctured surface. We work out the bracket on all generators when the surface is a thrice punctured sphere, or a trinion. The moduli space of convex real projective structures on a trinion, denoted by P,is a subset of X. Lastly, we determine explicit conditions in terms of C[X] that distinguish this moduli space.

math.AG↗
arXiv2014

Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups

Let $G$ be a real reductive algebraic group with maximal compact subgroup $K$, and let $F_r$ be a rank $r$ free group. We show that the space of closed orbits in $\mathrm{Hom}(F_r,G)/G$ admits a strong deformation retraction to the orbit space $\mathrm{Hom}(F_r,K)/K$. In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups $G$, such as $\mathrm{Sp}(4,\mathbb{R})$ and $\mathrm{U}(2,2)$. We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

math.AT↗
arXiv2014

E-polynomial of SL(2,C)-Character Varieties of Free groups

Let $\mathsf{F}_r$ be a free group of rank $r$, $\mathbb{F}_q$ a finite field of order q, and let $\mathrm{SL}_n(\mathbb{F}_q)$ act on $\mathrm{Hom}(\mathsf{F}_r, \mathrm{SL}_n(\mathbb{F}_q))$ by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits $\mathrm{Hom}(\mathsf{F}_r, \mathrm{SL}_n(\mathbb{F}_q))/\mathrm{SL}_n(\mathbb{F}_q)$. Our first main theorem is the implementation of this algorithm in the case $n=2$. As an application, we determine the $E$-polynomial of the character variety $\mathrm{Hom}(\mathsf{F}_r, \mathrm{SL}_2(\mathbb{C}))//!/\mathrm{SL}_2(\mathbb{C})$, and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.

math.AG↗
arXiv2013

Character Varieties and the Moduli of Quiver Representations

Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated groups, we show by collapsing arrows that their quotient spaces are equivalent. We then establish a general criterion for the moduli of G-valued quiver representations with relations to admit a strong deformation retraction to a compact quotient by pinching vertices on the quiver. This provides two different generalizations of main results in our previous work. Lastly, we establish quiver theoretic conditions for the moduli spaces of GL(n,C) and SL(n,C)-valued quiver representations to embed into traditional moduli spaces of quiver representations having constant dimension vector.

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arXiv2012

The Topology of Parabolic Character Varieties of Free Groups

Let G be a complex affine algebraic reductive group, and let K be a maximal compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or equal to 0, let X (respectively, Y) be the space of equivalence classes of representations of the free group of m+n generators in G (respectively, K) such that for each i between 1 and m, the image of the i-th free generator is conjugate to h_i. These spaces are parabolic analogues of character varieties of free groups. We prove that Y is a strong deformation retraction of X. In particular, X and Y are homotopy equivalent. We also describe explicit examples relating X to relative character varieties.

math.AG↗
arXiv2012

Singularities of free group character varieties

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating algebraic singularities to topological singularities, we show the moduli spaces X generally are not topological manifolds, except for a few examples we explicitly describe.

math.AG↗
arXiv2010

Computing SL(2,C) Central Functions with Spin Networks

Let G=SL(2,C) and F_r be a rank r free group. Given an admissible weight in N^{3r-3}, there exists a class function defined on Hom(F_r,G) called a central function. We show that these functions admit a combinatorial description in terms of graphs called trace diagrams. We then describe two algorithms (implemented in Mathematica) to compute these functions.

math.QA↗
arXiv2009

The topology of moduli spaces of free group representations

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the SL(3,C)-character variety of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character variety of a rank 3 free group is homotopic to a 6 sphere.

math.GN↗
arXiv2008

Spin networks and SL(2,C)-Character varieties

Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]^SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,C)-character variety of F_2 and a new proof of a classical result of Fricke, Klein, and Vogt.

math.QA↗
arXiv2008

Minimal Affine Coordinates for SL(3,C) Character Varieties of Free Groups

Let X be the moduli of SL(3,C) representations of a rank r free group. In this paper we determine minimal generators of the coordinate ring of X. This at once gives explicit global coordinates for the moduli and determines the dimension of the moduli's minimal affine embedding. Along the way, we utilize results concerning the moduli of r-tuples of matrices in gl(3,C). Consequently, we also state general invariant theoretic correspondences between the coordinate rings of the moduli of r-tuples of elements in gl(n,C), sl(n,C), and SL(n,C).

math.AG↗
arXiv2008

Obtaining the One-Holed Torus from Pants: Duality in an SL(3,C)-Character Variety

The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. The coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F imposed by the surface. In previous work, we have worked out the bracket on all generators when the surface is a three-holed sphere and when the surface is a one-holed torus. In this paper, we show how the symplectic leaves corresponding to these two different Poisson structures on X relate to each other. In particular, they are symplectically dual at a generic point. Moreover, the topological gluing map which turns the three-holed sphere into the one-holed torus induces a rank preserving Poisson map on C[X].

math.AG↗
arXiv2008

Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary

The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R by conjugation. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. We show the coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F imposed by the surface. Lastly, we work out the bracket on all generators when the surface is a three-holed sphere or a one-holed torus.

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