On $$A_1^2$$ A 1 2 restrictions of Weyl arrangements
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On A21 restrictions of Weyl arrangements Takuro Abe1 · Hiroaki Terao2 · Tan Nhat Tran2 Received: 14 May 2020 / Accepted: 1 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let A be a Weyl arrangement in an -dimensional Euclidean space. The freeness of restrictions of A was first settled by a case-by-case method by Orlik and Terao (Tôhoku Math J 52: 369–383, 1993), and later by a uniform argument by Douglass (Represent Theory 3: 444–456, 1999). Prior to this, Orlik and Solomon (Proc Symp Pure Math Amer Math Soc 40(2): 269–292, 1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik et al. (Adv Stud Pure Math 8: 461–77, 1986) asserts that the exponents of any A1 restriction, i.e., the restriction of A to a hyperplane, are given by {m 1 , . . . , m −1 }, where exp(A) = {m 1 , . . . , m } with m 1 ≤ · · · ≤ m . As a next step towards conceptual understanding of the restriction exponents, we will investigate the A21 restrictions, i.e., the restrictions of A to the subspaces of type A21 . In this paper, we give a combinatorial description of the exponents and describe bases for the modules of derivations of the A21 restrictions in terms of the classical notion of related roots by Kostant (Proc Nat Acad Sci USA 41:967–970, 1955). Keywords Root system · Weyl arrangement · Restriction · Freeness · Exponent · Basis Mathematics Subject Classification Primary 32S22; Secondary 17B22
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Tan Nhat Tran [email protected] Takuro Abe [email protected] Hiroaki Terao [email protected]
1
Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan
2
Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo 060-0810, Japan
123
Journal of Algebraic Combinatorics
1 Introduction Assume that V = R with the standard inner product (·, ·). Denote by an irreducible (crystallographic) root system in V and by + a positive system of . Let A be the Weyl arrangement of + . Denote by L(A) the intersection poset of A. For each X ∈ L(A), we write A X for the restriction of A to X . Set L p (A) := {X ∈ L(A) | codim(X ) = p} for 0 ≤ p ≤ . Let W be the Weyl group of and let m 1 , . . . , m with m 1 ≤ · · · ≤ m be the exponents of W . Notation 1.1 If X ∈ L p (A), then X := ∩ X ⊥ is a root system of rank p. A positive + system of X is taken to be + X := ∩ X . Let X be the base of X associated + with X . Definition 1.2 A subspace X ∈ L(A) is said to be of type T (or T for short) if X is a root system of type T . In this case, the restriction A X is said to be of type T (or T ). Weyl arrangements are important examples of free arrangements. In other words, the module D(A) of A-derivations is a free module. Furthermore, the exponents of A are the same as the exponents of W , i.e., exp(A) = {m 1 , . . . , m } (e.g., [23]). It is shown by Orlik and Solomon [19], using the classification of finite reflection groups, that the characteristic
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