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Lonardo Rabelo

Publications and source records attributed to Lonardo Rabelo.

7 recordsLinked to original sources
arXiv2026

Integral Homology and Poincaré Polynomials of classical and exceptional Real Flag Manifolds

This paper computes the integral homology of real flag manifolds associated with split real forms of classical and exceptional semisimple Lie algebras. Using the cellular homology provided by the Bruhat decomposition, we introduce a unified framework to systematically determine the coefficients of the boundary operator, explicitly resolving the issue of calculating their signs. This is achieved by computing the degree of change of coordinate maps between different reduced decompositions of Weyl group elements, analyzing commutation and braid relations through Lie bracket computations and exponential identities. By adopting the normal form of Weyl group elements as a canonical choice for reduced decompositions, we establish an explicit algorithmic implementation for these homology computations. As a direct application, we derive the Poincaré polynomials for the classical types $B_n, C_n$, and $D_n$ for $n \leqslant 7$, and for the exceptional types $F_4, E_6$, and $E_7$. With the aid of these polynomials, we address the question of the orientability of split real flag manifolds of exceptional Lie algebras.

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arXiv2025

The Bruhat Order on Symmetric Groups via Intrinsic Coverings of Compositions

Lehmer's code defines a bijection between the symmetric group and the set of staircase compositions. In this paper, we characterize a poset structure on these compositions that is equivalent to the strong Bruhat order on the symmetric group. This construction is intrinsic and does not require any reference to the associated permutations.

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arXiv2024

On the computation of homology of type A real flag manifolds

In this paper, we present a closed, computable formula for the cellular homology coefficients of real flag manifolds associated with split real forms of type A. We demonstrate the process using movements within the code diagram for permutations. Additionally, we compute the third and fourth homology groups and provide generators for the free part up to the sixth homology group.

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arXiv2020

Integral homology of real isotropic and odd orthogonal Grassmannians

We obtain a combinatorial expression for the coefficients of the boundary map of real isotropic and odd orthogonal Grassmannians providing a natural generalization of the formulas already obtained for Lagrangian and maximal isotropic Grassmannians. The results are given in terms of the classification into four types of covering pairs among the Schubert cells when identified with signed $k$-Grassmannian permutations. It turns out that these coefficients only depend on the positions changed over each pair of permutations. As an application, we give an orientability criterion, exhibit a symmetry of these coefficients and, compute low-dimensional homology groups.

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arXiv2018

Cellular Homology of Real Flag Manifolds

Let $\mathbb{F}_{Θ}=G/P_{Θ}$ be a generalized flag manifold, where $G$ is a real noncompact semi-simple Lie group and $P_{Θ}$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow $\mathbb{F}_Θ$ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in $\mathbb{R}^{n}$ and use them to compute the boundary operator $\partial $ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of $\partial $ are $0$ or $\pm 2$ (so that $\mathbb{Z}_{2}$-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse-Witten complex is solved.

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arXiv2018

Covering relations of k-Grassmannian permutations of type B

The main result of this work is the characterization of the covering relations of the Bruhat order of the maximal parabolic quotients of type B. Our approach is mainly combinatorial and is based in the pattern of the corresponding permutations also called signed $k$-Grassmannians permutations. We obtain that a covering relation can be classified in four different pairs of permutations. This answers a question raised by Ikeda and Matsumura providing a nice combinatorial model for maximal parabolic quotients of type B.

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