## Kallipoliti, Myrto and Tzanaki, Eleni - Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$

dmtcs:2495 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$

Authors: Kallipoliti, Myrto and Tzanaki, Eleni

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.

Source : oai:HAL:hal-01337790v1
Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Submitted on: November 21, 2016
Keywords: Shi hyperplane arrengements,abacus diagram,affine permutations,lattice paths,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]