Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$
In the present paper, the relation between the dominant regions in the $m$-Shi arrangement of types $B_n/C_n$, and those of the $m$-Shi arrangement of type $A_{n-1}$ is investigated. More precisely, it is shown explicitly how the sets $R^m(B_n)$ and $R^m(C_n)$, of dominant regions of the $m$-Shi arr...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2015-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2495/pdf |
| _version_ | 1797270248907341824 |
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| author | Myrto Kallipoliti Eleni Tzanaki |
| author_facet | Myrto Kallipoliti Eleni Tzanaki |
| author_sort | Myrto Kallipoliti |
| collection | DOAJ |
| description | In the present paper, the relation between the dominant regions in the $m$-Shi arrangement of types $B_n/C_n$, and those of the $m$-Shi arrangement of type $A_{n-1}$ is investigated. More precisely, it is shown explicitly how the sets $R^m(B_n)$ and $R^m(C_n)$, of dominant regions of the $m$-Shi arrangement of types $B_n$ and $C_n$ respectively, can be projected to the set $R^m(A_{n-1})$ of dominant regions of the $m$-Shi arrangement of type $A_{n-1}$. This is done by using two different viewpoints for the representative alcoves of these regions: the Shi tableaux and the abacus diagrams. Moreover, bijections between the sets $R^m(B_n)$, $R^m(C_n)$, and lattice paths inside a rectangle $n\times{mn}$ are provided. |
| first_indexed | 2024-04-25T02:01:16Z |
| format | Article |
| id | doaj.art-f0e4db0b2c9b4d11977910fd035661d0 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:16Z |
| publishDate | 2015-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-f0e4db0b2c9b4d11977910fd035661d02024-03-07T15:01:26ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502015-01-01DMTCS Proceedings, 27th...Proceedings10.46298/dmtcs.24952495Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$Myrto Kallipoliti0https://orcid.org/0000-0003-2188-6552Eleni Tzanaki1Fakultät für Mathematik [Wien]Department of Applied Mathematics [Heraklion]In the present paper, the relation between the dominant regions in the $m$-Shi arrangement of types $B_n/C_n$, and those of the $m$-Shi arrangement of type $A_{n-1}$ is investigated. More precisely, it is shown explicitly how the sets $R^m(B_n)$ and $R^m(C_n)$, of dominant regions of the $m$-Shi arrangement of types $B_n$ and $C_n$ respectively, can be projected to the set $R^m(A_{n-1})$ of dominant regions of the $m$-Shi arrangement of type $A_{n-1}$. This is done by using two different viewpoints for the representative alcoves of these regions: the Shi tableaux and the abacus diagrams. Moreover, bijections between the sets $R^m(B_n)$, $R^m(C_n)$, and lattice paths inside a rectangle $n\times{mn}$ are provided.https://dmtcs.episciences.org/2495/pdfshi hyperplane arrengementsabacus diagramaffine permutationslattice paths[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Myrto Kallipoliti Eleni Tzanaki Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ Discrete Mathematics & Theoretical Computer Science shi hyperplane arrengements abacus diagram affine permutations lattice paths [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ |
| title_full | Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ |
| title_fullStr | Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ |
| title_full_unstemmed | Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ |
| title_short | Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ |
| title_sort | bijections of dominant regions in the m shi arrangements of type a b and c |
| topic | shi hyperplane arrengements abacus diagram affine permutations lattice paths [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2495/pdf |
| work_keys_str_mv | AT myrtokallipoliti bijectionsofdominantregionsinthemshiarrangementsoftypeabandc AT elenitzanaki bijectionsofdominantregionsinthemshiarrangementsoftypeabandc |