On A12 restrictions of Weyl arrangements

Let A be a Weyl arrangement in an ℓ-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 { m₁, … , m_ℓ-1} , where exp (A) = { m₁, … , m_ℓ} with m₁≤ ⋯ ≤ m_ℓ. As a next step towards conceptual understanding of the restriction exponents, we will investigate the A12 restrictions, i.e., the restrictions of A to the subspaces of type A12. In this paper, we give a combinatorial description of the exponents and describe bases for the modules of derivations of the A12 restrictions in terms of the classical notion of related roots by Kostant (Proc Nat Acad Sci USA 41:967–970, 1955).