arXiv subjects

Ana Casimiro

Publications and source records attributed to Ana Casimiro.

7 recordsLinked to original sources

arXiv2026

(Equivariant) Cohomology of Homogeneous Spaces Associated with Composition Algebras

We study homogeneous spaces arising from embeddings of associative composition algebras of dimensions 2 and 4, with particular emphasis on the Hamiltonian quaternion algebra. For inclusions D \hookrightarrow C, we analyze the (equivariant) geometry and topology of the quotients GL(n,C)/GL(n,D). In the quaternionic case, we prove that these spaces are equivariantly diffeomorphic to homogeneous vector bundles over compact symmetric spaces, and hence admit equivariant deformation retractions onto compact homogeneous models. This description determines their homotopy type and reduces the computation of their cohomological invariants to the compact setting. Using this reduction together with Weyl group invariant theory, we compute the rational equivariant cohomology rings with respect to maximal tori and obtain explicit formulas for the Poincaré polynomials of the quaternionic homogeneous spaces. We further extend the analysis to noncompact symmetric spaces associated with Clifford groups. In this setting, we establish equivariant deformation retractions onto compact homogeneous spaces and derive explicit expressions for their Betti numbers.

math.AG↗

arXiv2023

Cayley-Menger extension of metrics on affine spaces

Associated to any affine space A endowed with a metric structure of arbitrary signature we consider the space of affine functionals operating on the space of quadratic functions of A. On this functional space we characterize a symmetric bilinear form derived from the metric structure in a functorial way. We explore the geometrical relations of the relevant objects in this new metric space. Their properties encode all characteristics known in the literature for euclidean squared distance matrices, Cayley-Menger matrices and determinants, squared distance coordinate systems, and Lie and Möbius sphere geometries. Birthing this form as Cayley-Menger product, it represents a geometrical foundation unifying results in all these areas, extending them to metric affine spaces or bundles.

math.MG↗

arXiv2023

Implicitization Of A Plane Curve Germ

Let $Y=\{f(x,y)=0\}$ be the germ of an irreducible plane curve. We present an algorithm to obtain polynomials, whose valuations coincide with the semigroup generators of $Y$. These polynomials are obtained sequentially, adding terms to the previous one in an appropriate way. To construct this algorithm, we perform truncations of the parametrization of $Y$ induced by the Puiseux Theorem. Then, an implicitization theorem of Tropical Geometry (Theorem $1.1$ of \cite{CM}) for plane curves is applied to the truncations. The identification between the local ring and the semigroup of $Y$ plays a key role in the construction of the algorithm, allowing us to carry out a formal elimination process, which we prove to be finite. The complexity of this elimination process equals the complexity of a integer linear programming problem. This algorithm also allows us to obtain an approximation of the series $f$, with the same multiplicity and characteristic exponents. We present a pseudocode of the algorithm and examples, where we compare computation times between an implementation of our algorithm and elimination through Gröbner bases.

math.AG↗

arXiv2022

Idempotent Varieties of Incidence Monoids and Bipartite Posets

The algebraic variety defined by the idempotents of an incidence monoid is investigated. Its irreducible components are determined. The intersection with an antichain submonoid is shown to be the union of these irreducible components. The antichain monoids of bipartite posets are shown to be orthodox semigroups. The Green's relations are explicitly determined, and applications to conjugacy problems are described. In particular, it is shown that two elements in the antichain monoid are primarily conjugate in the monoid if and only if they belong to the same $\mathcal{J}$-class and their multiplication by an idempotent of the same $\mathcal{J}$-class gives conjugate elements in the group.

math.CO↗

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

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↗

arXiv2012

Stability of Affine G-varieties and Irreducibility in Reductive Groups

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-Röhrle criteria, in the case $X=G^{N}$. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions.

math.RT↗