Geodesic
Manifolds of isospectral arrow matrices
Sbornik. Mathematics, Tome 212 (2021) no. 5, pp. 605-635 Cet article a éte moissonné depuis la source Math-Net.Ru

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An arrow matrix is a matrix with zeros outside the main diagonal, the first row and the first column. We consider the space $M_{\operatorname{St}_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove that this space is a smooth $2n$-manifold with a locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{\operatorname{St}_n,\lambda}/T^n$ is not a polytope, hence $M_{\operatorname{St}_n,\lambda}$ is not a quasitoric manifold. However, there is an action of a semidirect product $T^n\rtimes\Sigma_n$ on $M_{\operatorname{St}_n,\lambda}$, and the orbit space of this action is a certain simple polytope $\mathscr{B}^n$ obtained from the cube by cutting off codimension-2 faces. In the case $n=3$, the space $M_{\operatorname{St}_3,\lambda}/T^3$ is a solid torus with boundary subdivided into hexagons in a regular way. This description allows us to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold $M_{\operatorname{St}_3,\lambda}$ and another manifold, its twin. Bibliography: 32 titles.
Keywords: fundamental domain, codimension-2 face cuts.
Mots-clés : sparse matrix, group action, moment map 
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A. A. Ayzenberg; V. M. Buchstaber. Manifolds of isospectral arrow matrices. Sbornik. Mathematics, Tome 212 (2021) no. 5, pp. 605-635. http://geodesic.mathdoc.fr/item/SM_2021_212_5_a0/

[1] A. A. Ayzenberg, “Locally standard torus actions and sheaves over Buchsbaum posets”, Sb. Math., 208:9 (2017), 1261–1281 | DOI | DOI | MR | Zbl

[2] A. Ayzenberg, “Locally standard torus actions and $h'$-numbers of simplicial posets”, J. Math. Soc. Japan, 68:4 (2016), 1725–1745 | DOI | MR | Zbl

[3] A. Ayzenberg, “Homology cycles in manifolds with locally standard torus actions”, Homology, Homotopy Appl., 18:1 (2016), 1–23 | DOI | MR | Zbl

[4] A. Ayzenberg, “Topological model for $h''$-vectors of simplicial manifolds”, Bol. Soc. Mat. Mex. (3), 23:1 (2017), 413–421 | DOI | MR | Zbl

[5] A. Ayzenberg, “Space of isospectral periodic tridiagonal matrices”, Algebr. Geom. Topol., 20:6 (2020), 2957–2994 | DOI | MR

[6] A. A. Aizenberg, V. M. Buchstaber, “Nerve complexes and moment-angle spaces of convex polytopes”, Proc. Steklov Inst. Math., 275 (2011), 15–46 | DOI | MR | Zbl

[7] A. Ayzenberg, V. Buchstaber, “Manifolds of isospectral matrices and Hessenberg varieties”, Int. Math. Res. Not. IMRN, 2020, rnz388, 12 pp. | DOI

[8] A. Ayzenberg, M. Masuda, “Volume polynomials and duality algebras of multi-fans”, Arnold Math. J., 2:3 (2016), 329–381 | DOI | MR | Zbl

[9] A. Ayzenberg, M. Masuda, Seonjeong Park, Haozhi Zeng, “Cohomology of toric origami manifolds with acyclic proper faces”, J. Symplectic Geom., 15:3 (2017), 645–685 | DOI | MR | Zbl

[10] A. M. Bloch, H. Flaschka, T. Ratiu, “A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra”, Duke Math. J., 61:1 (1990), 41–65 | DOI | MR | Zbl

[11] V. M. Buchstaber, N. Yu. Erokhovets, “Fullerenes, polytopes and toric topology”, Combinatorial and toric homotopy, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 35, World Sci. Publ., Hackensack, NJ, 2018, 67–178 | DOI | MR | Zbl

[12] V. M. Buchstaber, I. Yu. Limonchenko, “Massey products, toric topology and combinatorics of polytopes”, Izv. Math., 83:6 (2019), 1081–1136 | DOI | DOI | MR | Zbl

[13] V. M. Buchstaber, T. E. Panov, Toric topology, Math. Surveys Monogr., 204, Amer. Math. Soc., Providence, RI, 2015, xiv+518 pp. | DOI | MR | Zbl

[14] V. M. Buchstaber, S. Terzić, “The foundations of $(2n,k)$-manifolds”, Sb. Math., 210:4 (2019), 508–549 | DOI | DOI | MR | Zbl

[15] A. Cannas da Silva, V. Guillemin, A. R. Pires, “Symplectic origami”, Int. Math. Res. Not. IMRN, 2011:18 (2011), 4252–4293 | DOI | MR | Zbl

[16] M. W. Davis, T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR | Zbl

[17] F. De Mari, M. Pedroni, “Toda flows and real Hessenberg manifolds”, J. Geom. Anal., 9:4 (1999), 607–625 | DOI | MR | Zbl

[18] S. Fisk, A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices, 2005, arXiv: math/0502408

[19] J. Grbić, A. Linton, “Lowest-degree triple Massey products in moment-angle complexes”, Russian Math. Surveys, 75:6 (2020), 1159–1161 | DOI | DOI | MR | Zbl

[20] V. Guillemin, L. Jeffrey, R. Sjamaar, “Symplectic implosion”, Transform. Groups, 7:2 (2002), 155–184 | DOI | MR | Zbl

[21] I. Krichever, K. L. Vaninsky, “The periodic and open Toda lattice”, Mirror symmetry, IV (Montreal, QC, 2000), AMS/IP Stud. Adv. Math., 33, Amer. Math. Soc., Providence, RI, 2002, 139–158 | MR | Zbl

[22] M. Masuda, T. Panov, “On the cohomology of torus manifolds”, Osaka J. Math., 43:3 (2006), 711–746 | MR | Zbl

[23] P. van Moerbeke, “The spectrum of Jacobi matrices”, Invent. Math., 37:1 (1976), 45–81 | DOI | MR | Zbl

[24] J. Moser, “Finitely many points on the line under the influence of an exponential potential – an integrable system”, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, WA, 1974), Lecture Notes in Phys., 38, Springer, Berlin, 1975, 467–497 | DOI | MR | Zbl

[25] T. Nanda, “Differential equations and the $QR$ algorithm”, SIAM J. Numer. Anal., 22:2 (1985), 310–321 | DOI | MR | Zbl

[26] I. Novik, E. Swartz, “Socles of Buchsbaum modules, complexes and posets”, Adv. Math., 222:6 (2009), 2059–2084 | DOI | MR | Zbl

[27] I. Novik, E. Swartz, “Gorenstein rings through face rings of manifolds”, Compos. Math., 145:4 (2009), 993–1000 | DOI | MR | Zbl

[28] G. Yu. Panina, “Cyclopermutohedron”, Proc. Steklov Inst. Math., 288 (2015), 132–144 | DOI | DOI | MR | Zbl

[29] G. A. Reisner, “Cohen–Macaulay quotients of polynomial rings”, Adv. Math., 21:1 (1976), 30–49 | DOI | MR | Zbl

[30] P. Schenzel, “On the number of faces of simplicial complexes and the purity of Frobenius”, Math. Z., 178:1 (1981), 125–142 | DOI | MR | Zbl

[31] R. P. Stanley, Combinatorics and commutative algebra, Progr. Math., 41, 2nd ed., Birkhäuser Boston Inc., Boston, MA, 1996, x+164 pp. | DOI | MR | Zbl

[32] C. Tomei, “The topology of isospectral manifolds of tridiagonal matrices”, Duke Math. J., 51:4 (1984), 981–996 | DOI | MR | Zbl