Year: 2009
Type: article
Abstract: We describe a software package for constructing minimal free resolutions of graded GL n (Q)-equivariant modules M over Q[x 1 , . . ., x n ] such that, for all i, the i-th syzygy module of M is generated in a single degree.We do so by describing some algorithms for manipulating polynomial representations of GL n (Q) following ideas of Olver and Eisenbud-Fløystad-Weyman.INTRODUCTION.This article describes the Macaulay 2 package PieriMaps, which defines maps of representations of the general linear group GL n (Q) of the formHere dominant weights of GL n (Q) are identified with weakly decreasing sequences λ of length n, and S λ (Q n ) denotes the irreducible representation of highest weight λ .Such maps are of general importance, and appeared recently in the work of Eisenbud, Fløystad and Weyman [EFW].The package also describes certain related maps in characteristic p.Let K be a field, and A := K[x 1 , . . ., x n ] the polynomial ring in n variables.In a recent paper of Eisenbud and Schreyer [ES], a theorem regarding the "shape" of minimal free resolutions of Cohen-Macaulay A-modules was established.Given a Cohen-Macaulay A-module M, the Betti diagram β i, j (M) is the number of generators of degree j in the ith syzygy module of a minimal free resolution of M. A Betti diagram is pure if for each i, β i, j (M) = 0 for at most one j.In this case, we let (d 1 , . . ., d r ) be the degree sequence of β : that is, β i,d i (M) = 0 for all i.The theorem mentioned above states that any Betti diagram of a Cohen-Macaulay module is a rational linear combination of pure Betti diagrams.The Herzog-Kühl equations [HK, Theorem 1] show that each strictly increasing degree sequence determines the corresponding Betti diagram up to a rational multiple.For char K = 0, Eisenbud, Fløystad, and Weyman [EFW] constructed pure free resolutions for each degree sequence which live in the category of GL n (K) representations.Eisenbud and Schreyer give a characteristic free construction which gives different rational multiples of the Betti diagrams in general.It is still not completely known which multiples can and cannot come from the Betti diagram of a graded module.For example, it is an interesting open problem to determine for a given degree sequence the smallest integer multiple given by the Herzog-Kühl equations which actually comes from a module.It is the goal of this article to describe how these resolutions can be represented concretely in Macaulay 2 [M2].One can find Z-forms for these maps and work in positive characteristic, and one such Z-form is implemented, but in general it will not produce pure resolutions.We should mention that this choice of Z-form is not unique.It would be interesting to investigate how often the characteristic p resolutions will be pure, and to construct Z-forms which give equivariant pure resolutions in positive characteristic.
Author: Steven V. Sam
Institution: Massachusetts Institute of Technology
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1983549949