  In 1977, Orlik--Randell construct a nice integral basis of the middle
homology group of the Milnor fiber associated to an invertible polynomial of
chain type and they conjectured that it is represented by a distinguished basis
of vanishing cycles. The purpose of this paper is to prove the algebraic
counterpart of the Orlik--Randell conjecture. Under the homological mirror
symmetry, we may expect that the triangulated category of maximally-graded
matrix factorizations for the Berglund--H\"{u}bsch transposed polynomial admits
a full exceptional collection with a nice numerical property. Indeed, we show
that the category admits a Lefschetz decomposition with respect to a
polarization in the sense of Kuznetsov--Smirnov, whose Euler matrix are
calculated in terms of the "zeta function" of the inverse of the polarization.
As a corollary, it turns out that the homological mirror symmetry holds at the
level of lattices, namely, the Grothendieck group of the category with the
Euler form is isomorphic to the middle homology group with the intersection
form (with a suitable sign).

Algebraic Geometry