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## Space of isospectral periodic tridiagonal matrices

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space of Hermitian periodic tridiagonal n×n-matrices with a fixed simple spectrum. Using discrete Shroedinger operator we give a condition on the spectrum which guarantees that this space is a manifold. The space carries a natural effective action of a compact (n−1)-torus. We describe the topology of its orbit space and, in particular, show that whenever the isospectral space is a manifold, its orbit space is homeomorphic to S4×Tn−3. The degenerate locus of the periodic Toda flow has combinatorial structure which can be described using the regular tiling of euclidean space by permutohedra. We study the topology of the isospectral space in vicinity of degenerate locus by methods of toric topology and algebraic combinatorics. Additive structure of homology and equivariant cohomology modules in vicinity of degenerate locus are computed in terms of the face ring of the special cell structure on a torus. We call this cell structure a wonderful cell subdivision due to its unique combinatorial properties.

Keywords: equivariant cohomologytorus actionface ringsimplicial posetToda flowmatrix spectrumisospectral spaceperiodic tridiagonal matrixdiscrete Schroedinger operatorpermutohedral tiling

Publication based on the results of:

Ayzenberg A., Algebraic and Geometric Topology 2020 Vol. 20 No. 6 P. 2957-2994

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized S\edt{c}hr\"{o}dinger operator we describe all spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The space $X_{n,\lambda}$ carries a natural effective action of a compact $(n-1)$-torus. ...

Added: January 14, 2020

Ayzenberg A., Masuda M., Orbit spaces of equivariantly formal torus actions / Cornell University. Series arXiv "math". 2019.

Let a compact torus T=T^{n−1} act on a smooth compact manifold X=X^{2n} effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If H^{odd}(X)=0 and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space Q=X/T is a homology (n+1)-sphere. If, in addition, π_1(X)=0, then Q is homeomorphic to S^{n+1}. ...

Added: January 14, 2020

Ayzenberg A., Бухштабер В. М., Manifolds of isospectral arrow matrices / . 2018. No. 10449.

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space MStn,λ of Hermitian arrow (n+1)×(n+1)-matrices with fixed simple spectrum λ. 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 ...

Added: October 15, 2018

Ayzenberg A., Masuda M., Park S. et al., Journal of Symplectic Geometry 2017 Vol. 15 No. 3 P. 645-685

A toric origami manifold is a generalization of a symplectic toric manifold (or a toric symplectic manifold). The origami symplectic form is allowed to degenerate in a good controllable way in contrast to the usual symplectic form. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes, and the cohomology and equivariant ...

Added: September 19, 2017

Ayzenberg A., Бухштабер В. М., Математический сборник 2021

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 this space to be a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe ...

Added: November 6, 2020

Gayfullin S., Journal of Algebra 2021 No. 573 P. 364-392

In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov. ...

Added: February 6, 2021

Finkelberg M., Braverman A., Shiraishi J., Providence : American Mathematical Society, 2014

Let G be an almost simple simply connected complex Lie group, and let G/U be its base affine space. In this paper we formulate a conjecture which provides a new geometric interpretation of the Macdonald polynomials associated to G via perverse coherent sheaves on the scheme of formal arcs in the affinizationof G/U. We prove ...

Added: March 5, 2015

Gayfullin S., Automorphisms of Danielewski varieties / Cornell University. Series arXiv "math". 2018. No. arXiv:1709.09237.

In 2007, Dubouloz introduced Danielewski varieties. Such varieties general- ize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov. ...

Added: September 1, 2018

Ayzenberg A., Arnold Mathematical Journal 2020 P. 1-24

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number 12dimX−dimT12dimX−dimT is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that HP2/T3≅S5HP2/T3≅S5 and S6/T2≅S4S6/T2≅S4, for the homogeneous spaces HP2=Sp(3)/(Sp(2)×Sp(1))HP2=Sp(3)/(Sp(2)×Sp(1)) and S6=G2/SU(3)S6=G2/SU(3). Here, the maximal tori of ...

Added: November 19, 2020

Arzhantsev I., Acta Arithmetica 2018 Vol. 186 No. 1 P. 87-99

We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwarz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwarz-Halphen's Theorem ...

Added: October 20, 2018

Ayzenberg A., Cherepanov V., Torus actions of complexity one in non-general position / Cornell University. Series arXiv "math". 2019. No. 1905.04761.

Let the compact torus Tn−1 act on a smooth compact manifold X2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X2n/Tn−1 if the action is cohomologically equivariantly formal (which essentially means that Hodd(X2n;Z)=0). It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite ...

Added: October 23, 2019

Ayzenberg A., Torus action on quaternionic projective plane and related spaces / Cornell University. Series arXiv "math". 2019. No. 1903.03460.

For an action of a compact torus T on a smooth compact manifold~X with isolated fixed points the number 12dimX−dimT is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that HP2/T3≅S5 and S6/T2≅S4, for the homogeneous spaces HP2=Sp(3)/(Sp(2)×Sp(1)) and S6=G2/SU(3). Here the maximal tori of the corresponding Lie ...

Added: October 23, 2019

Arzhantsev I., Gayfullin S., Mathematische Nachrichten 2017 Vol. 290 No. 5-6 P. 662-671

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of is pointed, the group is a finite extension of . As an application, ...

Added: February 19, 2017

Panov T., Dolbeault cohomology of complex manifolds with torus action / Cornell University. Series arXiv "math". 2019.

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga model for the ordinary Dolbeault cohomology algebra. ...

Added: November 1, 2019

Zaitseva Y., Математические заметки 2019 Т. 105 № 6 С. 824-838

В работе получено описание однородных локально нильпотентных дифференцирований алгебры регулярных функций некоторого класса триномиальных гиперповерхностей. Данный класс включает в себя все нефакториальные триномиальные гиперповерхности. ...

Added: September 19, 2019

Arzhantsev I., Ricerche di Matematica 2021

We show that an effective action of the one-dimensional torus G_m on a normal affine algebraic variety X can be extended to an effective action of a semi-direct product G_m⋌G_a with the same general orbit closures if and only if there is a divisor D on X that consists of G_m-fixed points. This result is applied to the study of orbits of the automorphism group Aut(X) on X. ...

Added: August 16, 2021

Ayzenberg A., Cherepanov V., Osaka Journal of Mathematics 2021 Vol. 58 No. 4 P. 839-853

Let the compact torus Tn1 act on a smooth compact manifold X2n eectively with nonempty nite set of xed points. We pose the question: what can be said
about the orbit space X2n{Tn1 if the action is cohomologically equivariantly formal
(which essentially means that HoddpX2n;Zq 0)? It happens that homology of the orbit
space can be arbitrary ...

Added: October 31, 2019

Panov T., Ishida H., Basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds / Cornell University. Series arXiv "math". 2018.

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a ...

Added: November 1, 2019

Ayzenberg Anton, Buchstaber V.M., International Mathematical Research Notices 2021 Vol. 2021 No. 21 P. 16671-16692

We study the space Xh of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that Xh is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of ...

Added: June 16, 2021

Buchstaber V.M., Terzić S., Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 397-463

The family of the complex Grassmann manifolds G(n,k) with the canonical action of the torus T-n = T-n and the analogue of the moment map mu: G(n,k) ->Delta(n,)(k) for the hypersimplex Delta(n,) (k), is well known. In this paper we study the structure of the orbit space G(n,k)/T-n by developing the methods of toric geometry ...

Added: June 18, 2021

Arzhantsev I., Liendo A., Stasyuk T., Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499

Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...

Added: July 29, 2020

Ayzenberg A., Труды Математического института им. В.А. Стеклова РАН 2018 Т. 302 С. 23-40

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated xed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a ltration of the orbit manifold by orbit dimensions. The subset ...

Added: October 15, 2018

Arzhantsev I., Hausen J., Mathematical Research Letters 2007 Vol. 14 No. 1 P. 129-136

Given a multigraded algebra A, it is a natural question whether or not for
two homogeneous components A_u and A_v, the product A_nuA_nv is the whole component
A_nu+nv for n big enough. We give combinatorial and geometric answers to this question. ...

Added: July 10, 2014

Gayfullin S., Zaitseva Y., Journal of Algebra and its Applications 2019 Vol. 18 No. 10 P. 1950196-1-1950196-14

We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras. ...

Added: September 10, 2019