TY - JOUR
T1 - The shi arrangement and the Ish arrangement
AU - Armstrong, Drew
AU - Rhoades, Brendon
PY - 2012
Y1 - 2012
N2 - This paper is about two arrangements of hyperplanes. The first - the Shi arrangement - was introduced by Jian-Yi Shi (1986) to describe the Kazhdan-Lusztig cells in the affine Weyl group of type A. The second - the Ish arrangement - was recently defined by the first author, who used the two arrangements together to give a new interpretation of the q, t-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with c "ceilings" and d "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.
AB - This paper is about two arrangements of hyperplanes. The first - the Shi arrangement - was introduced by Jian-Yi Shi (1986) to describe the Kazhdan-Lusztig cells in the affine Weyl group of type A. The second - the Ish arrangement - was recently defined by the first author, who used the two arrangements together to give a new interpretation of the q, t-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with c "ceilings" and d "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.
UR - http://www.scopus.com/inward/record.url?scp=82755171905&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=82755171905&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-2011-05521-2
DO - 10.1090/S0002-9947-2011-05521-2
M3 - Article
AN - SCOPUS:82755171905
VL - 364
SP - 1509
EP - 1528
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 3
ER -