The freeness of Ish arrangements. (English. French summary) Zbl 1335.05189
Proceedings of the 27th international conference on formal power series and algebraic combinatorics, FPSAC 2015, Daejeon, South Korea, July 6–10, 2015. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 273-284 (2015).
Summary: The Ish arrangement was introduced by D. Armstrong [J. Comb. 4, No. 2, 157–190 (2013; Zbl 1275.05061)] to give a new interpretation of the \(q,t\)-Catalan numbers of Garsia and Haiman. D. Armstrong and B. Rhoades [Trans. Am. Math. Soc. 364, No. 3, 1509–1528 (2012; Zbl 1238.05271)] 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.
For the entire collection see [Zbl 1333.05004].
For the entire collection see [Zbl 1333.05004].
MSC:
| 05E10 | Combinatorial aspects of representation theory |
| 05A15 | Exact enumeration problems, generating functions |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
