Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015)
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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.
@article{DMTCS_2015_special_285_a39,
author = {Kallipoliti, Myrto and Tzanaki, Eleni},
title = {Bijections of dominant regions in the $m${-Shi} arrangements of type $A$, $B$ and $C$},
journal = {Discrete mathematics & theoretical computer science},
year = {2015},
volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
doi = {10.46298/dmtcs.2495},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2495/}
}
TY - JOUR AU - Kallipoliti, Myrto AU - Tzanaki, Eleni TI - Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ JO - Discrete mathematics & theoretical computer science PY - 2015 VL - DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) UR - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2495/ DO - 10.46298/dmtcs.2495 LA - en ID - DMTCS_2015_special_285_a39 ER -
%0 Journal Article %A Kallipoliti, Myrto %A Tzanaki, Eleni %T Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ %J Discrete mathematics & theoretical computer science %D 2015 %V DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) %U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2495/ %R 10.46298/dmtcs.2495 %G en %F DMTCS_2015_special_285_a39
Kallipoliti, Myrto; Tzanaki, Eleni. Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi: 10.46298/dmtcs.2495
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