Face counting formula for toric arrangements defined by root systems
A toric arrangement is a finite collection of codimension-1 subtori in a torus. The intersections of these subtori stratify the ambient torus into faces of various dimensions. The incidence relations among these faces give rise to many interesting combinatorial problems. One such problem is to obtai...
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2019-04-01
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| Series: | AKCE International Journal of Graphs and Combinatorics |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1016/j.akcej.2018.07.002 |
| _version_ | 1818328409406701568 |
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| author | Priyavrat Deshpande Kavita Sutar |
| author_facet | Priyavrat Deshpande Kavita Sutar |
| author_sort | Priyavrat Deshpande |
| collection | DOAJ |
| description | A toric arrangement is a finite collection of codimension-1 subtori in a torus. The intersections of these subtori stratify the ambient torus into faces of various dimensions. The incidence relations among these faces give rise to many interesting combinatorial problems. One such problem is to obtain a closed-form formula for the number of faces in terms of the intrinsic arrangement data. In this paper we focus on toric arrangements defined by crystallographic root systems. Such an arrangement is equipped with an action of the associated Weyl group. The main result is a formula that expresses the face numbers in terms of a sum of indices of certain subgroups of this Weyl group. |
| first_indexed | 2024-12-13T12:31:42Z |
| format | Article |
| id | doaj.art-bdd4c4e897ea457eb31cf29fb6d40e0d |
| institution | Directory Open Access Journal |
| issn | 0972-8600 |
| language | English |
| last_indexed | 2024-12-13T12:31:42Z |
| publishDate | 2019-04-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | AKCE International Journal of Graphs and Combinatorics |
| spelling | doaj.art-bdd4c4e897ea457eb31cf29fb6d40e0d2022-12-21T23:46:00ZengTaylor & Francis GroupAKCE International Journal of Graphs and Combinatorics0972-86002019-04-01161667710.1016/j.akcej.2018.07.00212092535Face counting formula for toric arrangements defined by root systemsPriyavrat Deshpande0Kavita Sutar1 Chennai Mathematical Institute Chennai Mathematical InstituteA toric arrangement is a finite collection of codimension-1 subtori in a torus. The intersections of these subtori stratify the ambient torus into faces of various dimensions. The incidence relations among these faces give rise to many interesting combinatorial problems. One such problem is to obtain a closed-form formula for the number of faces in terms of the intrinsic arrangement data. In this paper we focus on toric arrangements defined by crystallographic root systems. Such an arrangement is equipped with an action of the associated Weyl group. The main result is a formula that expresses the face numbers in terms of a sum of indices of certain subgroups of this Weyl group.http://dx.doi.org/10.1016/j.akcej.2018.07.002toric arrangementsface enumerations-vectoraffine weyl groups |
| spellingShingle | Priyavrat Deshpande Kavita Sutar Face counting formula for toric arrangements defined by root systems AKCE International Journal of Graphs and Combinatorics toric arrangements face enumerations -vector affine weyl groups |
| title | Face counting formula for toric arrangements defined by root systems |
| title_full | Face counting formula for toric arrangements defined by root systems |
| title_fullStr | Face counting formula for toric arrangements defined by root systems |
| title_full_unstemmed | Face counting formula for toric arrangements defined by root systems |
| title_short | Face counting formula for toric arrangements defined by root systems |
| title_sort | face counting formula for toric arrangements defined by root systems |
| topic | toric arrangements face enumerations -vector affine weyl groups |
| url | http://dx.doi.org/10.1016/j.akcej.2018.07.002 |
| work_keys_str_mv | AT priyavratdeshpande facecountingformulafortoricarrangementsdefinedbyrootsystems AT kavitasutar facecountingformulafortoricarrangementsdefinedbyrootsystems |