TY - JOUR
T1 - The freeness of Ish arrangements
AU - Abe, Takuro
AU - Suyama, Daisuke
AU - Tsujie, Shuhei
N1 - Funding Information:
The authors are grateful to Brendon Rhoades for suggesting a base chamber for the wall-crossing formula of the original Ish arrangement . The proof of Theorem 3.3 is based on his idea. We would also like to thank the referees for careful reading of our manuscript and for giving useful comments. The first author is partially supported by JSPS Grants-in-Aid for Young Scientists (B) No. 24740012 .
Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2017/2/1
Y1 - 2017/2/1
N2 - The Ish arrangement was introduced by Armstrong to give a new interpretation of the q,t-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free.
AB - The Ish arrangement was introduced by Armstrong to give a new interpretation of the q,t-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free.
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U2 - 10.1016/j.jcta.2016.09.008
DO - 10.1016/j.jcta.2016.09.008
M3 - Article
AN - SCOPUS:84991325406
SN - 0097-3165
VL - 146
SP - 169
EP - 183
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
ER -