1.1 Boij–Söderberg theory. Our original motivation comes from trying to construct pure free resolutions over the homogeneous coordinate ring of a quadric hypersurface. First we review some definitions for polynomial rings. Let A = K[x1,...,xn] be a polynomial ring with the standard grading. Given a finitely generated graded A-module M, its tor modules are naturally graded, and we set βi,j(M) = dimTor A i (M,K)j. A module M has a pure free resolution if Tor A i (M,K), when nonzero, is concentrated in a single degree di(M) for all i (sometimes one imposes Cohen–Macaulayness of M). In this case, d(M) = (d0(M),d1(M),...) is the degree sequence of M. Note that d0(M) < d1(M) < ···. Theorem 1.1 (Eisenbud–Fløystad–Weyman [EFW], Eisenbud–Schreyer [ES]). Given d0 < d1 < ·· · < dr (with r ≤ n), there is a module M with pure free resolution such that di(M) = di. One can make the same definition for quadric hypersurfaces B = K[x1,...,xn]/q(x) where q(x) is a homogeneous quadric. Note that minimal free resolutions over B are generally infinite in length, but become periodic of period 2 after n steps. Our proposed approach is to transform the resolutions of Eisenbud–Fløystad–Weyman. The basic idea is the following. Write A = Sym(V), which naturally has an action of G = GL(V). The polynomial representations Sλ(V) of G are parametrized by partitions λ with at most n parts. We pictorially represent the free module Sλ(V)⊗A as a Young diagram with λi boxes in the ith column. An example of an EFW complex for n = 4 i
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