82 research outputs found

### Hyperelliptic addition law

We construct an explicit form of the addition law for hyperelliptic Abelian
vector functions $\wp$ and $\wp'$. The functions $\wp$ and $\wp'$ 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 $\wp$ and $\wp'$.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 $M_{St_n,\lambda}$ of Hermitian arrow
$(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove
that this space is a smooth $2n$-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
$n\geqslant 3$, the orbit space $M_{St_n,\lambda}/T^n$ is not a polytope, hence
this manifold is not quasitoric. However, there is a natural permutation action
on $M_{St_n,\lambda}$ which induces the combined action of a semidirect product
$T^n\rtimes\Sigma_n$. The orbit space of this large action is a simple
polytope. The structure of this polytope is described in the paper.
In case $n=3$, the space $M_{St_3,\lambda}/T^3$ 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 $M_{St_3,\lambda}$ using the general theory developed by
the first author. This theory is also applied to a certain $6$-dimensional
manifold called the twin of $M_{St_3,\lambda}$. 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

In this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
$\mathcal{P}$ generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product $\times$ and a
join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative
associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb
Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative
threegraded ring of polynomials. The ring $\mathcal{RP}$ has the structure of a
graded Hopf algebra. It turns out that $\mathcal{P}$ has a natural Hopf
comodule structure over $\mathcal{RP}$. Faces operators $d_k$ that send a
polytope to the sum of all its $(n-k)$-dimensional faces define on both rings
the Hopf module structures over the universal Leibnitz-Hopf algebra
$\mathcal{Z}$. This structure gives a ring homomorphism \R\to\Qs\otimes\R,
where $\R$ is $\mathcal{P}$ or $\mathcal{RP}$. Composing this homomorphism with
the characters $P^n\to\alpha^n$ of $\mathcal{P}$, $P^n\to\alpha^{n+1}$ of
$\mathcal{RP}$, 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 $F$ 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 $\mathbb Q$, and find the
relations between the images, the homomorphisms and the Hopf comodule
structures. For each homomorphism $f,\;f_{\mathcal{RP}}$, and \F the images
of two polytopes coincide if and only if they have equal flag $f$-vectors.
Therefore algebraic structures on the images give the information about flag
$f$-vectors of polytopes.Comment: 61 page

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