80 research outputs found
Hyperelliptic addition law
We construct an explicit form of the addition law for hyperelliptic Abelian
vector functions and . The functions and 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 and .Comment: 18 pages, amslate
Manifolds of isospectral arrow matrices
An arrow matrix is a matrix with zeroes outside the main diagonal, first row,
and first column. We consider the space of Hermitian arrow
-matrices with fixed simple spectrum . We prove
that this space is a smooth -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
, the orbit space is not a polytope, hence
this manifold is not quasitoric. However, there is a natural permutation action
on which induces the combined action of a semidirect product
. The orbit space of this large action is a simple
polytope. The structure of this polytope is described in the paper.
In case , the space 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 using the general theory developed by
the first author. This theory is also applied to a certain -dimensional
manifold called the twin of . 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
In this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product and a
join of polytopes. is a commutative
associative bigraded ring of polynomials, and is a commutative associative
threegraded ring of polynomials. The ring has the structure of a
graded Hopf algebra. It turns out that has a natural Hopf
comodule structure over . Faces operators that send a
polytope to the sum of all its -dimensional faces define on both rings
the Hopf module structures over the universal Leibnitz-Hopf algebra
. This structure gives a ring homomorphism \R\to\Qs\otimes\R,
where is or . Composing this homomorphism with
the characters of , of
, 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 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 , and find the
relations between the images, the homomorphisms and the Hopf comodule
structures. For each homomorphism , and \F the images
of two polytopes coincide if and only if they have equal flag -vectors.
Therefore algebraic structures on the images give the information about flag
-vectors of polytopes.Comment: 61 page
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