The Zero Loci of F -triangles
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The Zero Loci of F -triangles Kyoji Saito1
Received: 15 July 2016 / Accepted: 1 December 2016 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017
Abstract We are interested in the zero locus of a Chapoton’s F -triangle as a polynomial in two real variables x and y. An expectation is that (1) the F -triangle of rank l as a polynomial in x for each fixed y ∈ [0, 1] has exactly l distinct real roots in [0, 1], and (2) ith root xi (y) (1 ≤ i ≤ l) as a function on y ∈ [0, 1] is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of F -triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above expectation is true. We formulate inductive conjectures and questions for further rank cases. This study gives a new insight on the zero loci of f + - and f -polynomials. Keywords F -triangle · A-triangle · Polyhedral cone · Real zero loci Mathematics Subject Classification (2010) 05A99
1 Chapoton’s F -triangle We recall the F -triangle F by F. Chapoton ([5]) associated with a complete simplicial fan (), called a cluster fan, associated with a finite root system , and introduced by Fomin and Zelevinsky [7, 8].1 Let be a finite (not necessary irreducible) root system of rank l, + ⊂ be a positive root system, and = {αi }i∈I ⊂ + be the associated simple basis where I is the
1 In
order to adjust to the study in the present note, the definition (1.1) changes its sign of variables from the original definition of Chapoton. One should be cautious that this change causes several sign changes in the sequel (e.g., (1.2), Proposition 4., (4.3), . . . etc.).
Kyoji Saito
[email protected] 1
Kavli Institute for the Physics and Mathematics of the Universe (WPI), the University of Tokyo, Tokyo, Japan
K. Saito
index set of order l. We identify I with associated Dynkin diagram whose underlying set is I . Let (J ) denotes the finite root system associated with the full sub-diagram J ⊂ I . A symmetric compatiblity relation (αβ) = 0 for α, β ∈ ≥−1 := + ∪ {−} was introduced in [9]: Theorem ([9, Theorem 1.10]) The cones spanned by sets of mutually compatible elements in ≥−1 define a complete simplicial fan Δ(Φ). Taking the distinction between positive and negative vertices − of () in account, F. Chapoton ([5, (1)]) introduced a refinement of the face counting generating function, called the F -triangle (see Footnote 1). Definition 1.1 For a root system , the F -triangle is l l fk,m (−x)k (−y)m , F () = F (x, y) =
(1.1)
k=0 m=0
where fk,m is the cardinality of the set of simplicial cones of () spanned by exactly k positive roots and m negative simple roots. The coefficient fk,m vanishes if k + m > l, hence the name triangle. For a formal convenience, we include an “empty root system” (∅) in the discussion. In that case, we set F ((∅)) := 1. The next and the most basic F -triangle is the case of rank 1, whi
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