dorsal/arxiv
View SchemaToda lattices with indefinite metric II: Topology of the iso-spectral manifolds
| Authors | Yuji Kodama, Jian Ye |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9609001 |
| URL | https://arxiv.org/abs/solv-int/9609001 |
Abstract
We consider the iso-spectral real manifolds of tridiagonal Hessenberg matrices with real eigenvalues. The manifolds are described by the iso-spectral flows of indefinite Toda lattice equations introduced by the authors [Physica, 91D (1996), 321-339]. These Toda lattices consist of $2^{N-1}$ different systems with hamiltonians $H = (1/2) \sum_{k=1}^{N} y_k^2 + \sum_{k=1}^{N-1} s_ks_{k+1} \exp(x_k-x_{k+1})$, where $s_i=\pm 1$. We compactify the manifolds by adding infinities according to the Toda flows which blow up in finite time except the case with all $s_is_{i+1}=1$. The resulting manifolds are shown to be nonorientable for $N>2$, and the symmetric group is the semi-direct product of $(\ZZ_2)^{N-1}$ and the permutation group $S_N$. These properties identify themselves with ``small covers'' introduced by Davis and Januszkiewicz [Duke Mathematical Journal, 62 (1991), 417-451]. As a corollary of our construction, we give a formula on the total numbers of zeroes for a system of exponential polynomials generated as Hankel determinant.
{
"annotation_id": "3b37e63f-8a75-475d-92bb-ef1577fd43bb",
"date_created": "2026-03-02T18:02:51.575000Z",
"date_modified": "2026-03-02T18:02:51.575000Z",
"file_hash": "e9f5dd9996bfb38e323faede7739057b31bd1f8bb2692c0c71d350a18ea74faf",
"private": false,
"record": {
"abstract": "We consider the iso-spectral real manifolds of tridiagonal Hessenberg\nmatrices with real eigenvalues. The manifolds are described by the iso-spectral\nflows of indefinite Toda lattice equations introduced by the authors [Physica,\n91D (1996), 321-339]. These Toda lattices consist of $2^{N-1}$ different\nsystems with hamiltonians $H = (1/2) \\sum_{k=1}^{N} y_k^2 + \\sum_{k=1}^{N-1}\ns_ks_{k+1} \\exp(x_k-x_{k+1})$, where $s_i=\\pm 1$. We compactify the manifolds\nby adding infinities according to the Toda flows which blow up in finite time\nexcept the case with all $s_is_{i+1}=1$. The resulting manifolds are shown to\nbe nonorientable for $N\u003e2$, and the symmetric group is the semi-direct product\nof $(\\ZZ_2)^{N-1}$ and the permutation group $S_N$. These properties identify\nthemselves with ``small covers\u0027\u0027 introduced by Davis and Januszkiewicz [Duke\nMathematical Journal, 62 (1991), 417-451]. As a corollary of our construction,\nwe give a formula on the total numbers of zeroes for a system of exponential\npolynomials generated as Hankel determinant.",
"arxiv_id": "solv-int/9609001",
"authors": [
"Yuji Kodama",
"Jian Ye"
],
"categories": [
"solv-int",
"hep-th",
"nlin.SI"
],
"title": "Toda lattices with indefinite metric II: Topology of the iso-spectral manifolds",
"url": "https://arxiv.org/abs/solv-int/9609001"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "f3b94f40-60b6-4a76-8f2a-fbd48f21da5a",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}