Characteristic quasi-polynomials of ideals and signed graphs of classical root systems

Abstract

With a main tool is signed graphs, we give a full description of the characteristic quasi-polynomials of ideals of classical root systems (ABCD) with respect to the integer and root lattices. As a result, we obtain a full description of the characteristic polynomials of the toric arrangements defined by these ideals. As an application, we provide a combinatorial verification to the fact that the characteristic polynomial of every ideal subarrangement factors over the dual partition of the ideal in the classical cases.

Inductive and divisional posets

“…Furthermore, we give an explicit description of the exponents of $$ \mathcal {A}_{{\mathcal {I}}}$$ derived from an explicit induction table. This description turns out to be equivalent to the ones in [32]. We also give a characterization for supersolvability of $$\mathcal {A}_{{\Phi ^+} }$$ when $$\Phi$$ is of type $$B$$ (Theorem 7.17).…”

“…Furthermore, we give an explicit description of the exponents of 𝒜  derived from an explicit induction table. This description turns out to be equivalent to the ones in [32]. We also give a characterization for supersolvability of 𝒜 Φ + when Φ is of type 𝐵 (Theorem 7.17 The proof for the type 𝐴 case in Theorem 1.4 is a simple consequence of Theorem 7.2, which we give below.…”

Eulerian polynomials for subarrangements of Weyl arrangements

“…Enumerating the cardinality of the complement of A(Z q ) produces a quasi-polynomial, the characteristic quasi-polynomial χ quasi A (q) of A [KTT08]. This single quasi-polynomial encodes a number of combinatorial and topological information of several types of arrangements and has generated increasing interest recently (e.g., [CW12,BM14,Yos18a,Yos18b,TY19,Tra19]). Among the others, χ quasi A (q) has the first constituent identical with the characteristic polynomial χ A(R) (t) of A(R) which justified its name (e.g., [Ath96,KTT08]), and the last constituent identical with the characteristic polynomial χ A(S 1 ) (t) of A(S 1 ) [LTY,TY19].…”

“…Some partial results are known. If the root system Φ is of classical type and Ψ is an ideal of Φ + , then χ quasi Ψ (q) can be computed from information of the signed graph associated with Ψ [Tra19]. A result due to [Yos18b] applied to any root system, asserts that χ quasi ∅ (q) (or simply q ℓ ) can be written in terms of the Lam-Postnikov's Eulerian polynomial [LP18], shift operator, and…”

Period collapse in characteristic quasi-polynomials of hyperplane arrangements

“…Determining the minimum period of a quasi-polynomial is natural but in general a challenging problem. Since their introduction, the methods above motivated several discussions on the minimum period of the characteristic quasi-polynomial in the central case, e.g., [KTT11,CW12,Tra20], as well as help to derive many computational results, e.g., [KTT10,Tra19] that support the equality between the minimum period and the lcm period. However, to date, no proof has been known.…”