Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof)
Motivated by applications to perverse sheaves, we study the combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decompo...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210786 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof). Mikhail Kapranov and Vadim Schechtman. SIGMA 16 (2020), 062, 22 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862631851827396608 |
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| author | Kapranov, Mikhail Schechtman, Vadim |
| author_facet | Kapranov, Mikhail Schechtman, Vadim |
| citation_txt | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof). Mikhail Kapranov and Vadim Schechtman. SIGMA 16 (2020), 062, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Motivated by applications to perverse sheaves, we study the combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the ''contingency decomposition'' to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system Aₙ) introduced by T.K. Petersen. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the ''meta-matrix'' formed by the numbers of contingency matrices of various sizes is totally positive.
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| first_indexed | 2026-03-14T19:33:24Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210786 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T19:33:24Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
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| spelling | Kapranov, Mikhail Schechtman, Vadim 2025-12-17T14:39:17Z 2020 Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof). Mikhail Kapranov and Vadim Schechtman. SIGMA 16 (2020), 062, 22 pages 1815-0659 2020 Mathematics Subject Classification: 57Q05; 52B70 arXiv:1909.09793 https://nasplib.isofts.kiev.ua/handle/123456789/210786 https://doi.org/10.3842/SIGMA.2020.062 Motivated by applications to perverse sheaves, we study the combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the ''contingency decomposition'' to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system Aₙ) introduced by T.K. Petersen. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the ''meta-matrix'' formed by the numbers of contingency matrices of various sizes is totally positive. We are happy to dedicate this paper to Dmitry Borisovich Fuchs. Among several wonderful things he has done in mathematics, he is one of the pioneers in the study of cellular decompositions for symmetric products. We are grateful to Pavel Etingof for valuable discussions and for agreeing to include his work as an appendix to our paper. V.S. is grateful to the organizers of a conference in Zurich in August 2019, where he had a chance to meet P.E. We would like to thank Sergei Fomin for pointing out the relevance of the paper [17] to our work and for pointing out several misprints in the earlier version. We are also grateful to the referees for their useful remarks and corrections. The research of M.K. was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof) Article published earlier |
| spellingShingle | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof) Kapranov, Mikhail Schechtman, Vadim |
| title | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof) |
| title_full | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof) |
| title_fullStr | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof) |
| title_full_unstemmed | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof) |
| title_short | Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof) |
| title_sort | contingency tables with variable margins (with an appendix by pavel etingof) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210786 |
| work_keys_str_mv | AT kapranovmikhail contingencytableswithvariablemarginswithanappendixbypaveletingof AT schechtmanvadim contingencytableswithvariablemarginswithanappendixbypaveletingof |