82 research outputs found

    Hyperelliptic addition law

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    We construct an explicit form of the addition law for hyperelliptic Abelian vector functions β„˜\wp and β„˜β€²\wp'. The functions β„˜\wp and β„˜β€²\wp' form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of β„˜\wp and β„˜β€²\wp'.Comment: 18 pages, amslate

    Manifolds of isospectral arrow matrices

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    An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space MStn,Ξ»M_{St_n,\lambda} of Hermitian arrow (n+1)Γ—(n+1)(n+1)\times (n+1)-matrices with fixed simple spectrum Ξ»\lambda. We prove that this space is a smooth 2n2n-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If nβ©Ύ3n\geqslant 3, the orbit space MStn,Ξ»/TnM_{St_n,\lambda}/T^n is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on MStn,Ξ»M_{St_n,\lambda} which induces the combined action of a semidirect product Tnβ‹ŠΞ£nT^n\rtimes\Sigma_n. The orbit space of this large action is a simple polytope. The structure of this polytope is described in the paper. In case n=3n=3, the space MSt3,Ξ»/T3M_{St_3,\lambda}/T^3 is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold MSt3,Ξ»M_{St_3,\lambda} using the general theory developed by the first author. This theory is also applied to a certain 66-dimensional manifold called the twin of MSt3,Ξ»M_{St_3,\lambda}. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.Comment: 29 pages, 8 figure

    Polytopes, Hopf algebras and Quasi-symmetric functions

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    In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group P\mathcal{P} generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product Γ—\times and a join ⋇\divideontimes of polytopes. (P,Γ—)(\mathcal{P},\times) is a commutative associative bigraded ring of polynomials, and RP=(Zβˆ…βŠ•P,⋇)\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes) is a commutative associative threegraded ring of polynomials. The ring RP\mathcal{RP} has the structure of a graded Hopf algebra. It turns out that P\mathcal{P} has a natural Hopf comodule structure over RP\mathcal{RP}. Faces operators dkd_k that send a polytope to the sum of all its (nβˆ’k)(n-k)-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra Z\mathcal{Z}. This structure gives a ring homomorphism \R\to\Qs\otimes\R, where R\R is P\mathcal{P} or RP\mathcal{RP}. Composing this homomorphism with the characters Pnβ†’Ξ±nP^n\to\alpha^n of P\mathcal{P}, Pnβ†’Ξ±n+1P^n\to\alpha^{n+1} of RP\mathcal{RP}, and with the counit we obtain the ring homomorphisms f\colon\mathcal{P}\to\Qs[\alpha], f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha], and \F^*:\mathcal{RP}\to\Qs, where FF is the Ehrenborg transformation. We describe the images of these homomorphisms in terms of functional equations, prove that these images are rings of polynomials over Q\mathbb Q, and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism f,β€…β€ŠfRPf,\;f_{\mathcal{RP}}, and \F the images of two polytopes coincide if and only if they have equal flag ff-vectors. Therefore algebraic structures on the images give the information about flag ff-vectors of polytopes.Comment: 61 page
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