关于使用计算机代数软件计算Auslander Reiten箭图的技术咨询

Hey there! Awesome question—computing Auslander-Reiten (AR) quivers for path algebras and digging into related structures like exterior algebras is totally feasible with a handful of specialized tools. Here are the top options I’d recommend, along with how they can help you out:

• GAP + QPA Package
  QPA (Quivers, Path Algebras and Representations) is the industry standard for anyone working with path algebras and their representations. It can directly compute AR quivers for finite-dimensional path algebras over fields, and it supports a ton of operations to explore module structures—including those tied to exterior algebras, as long as you set up the algebra correctly.

  To get started, you’d first define your quiver, build the path algebra, then use commands like AuslanderReitenQuiver(your_algebra) to generate the AR quiver. You can also inspect individual modules, irreducible morphisms, Ext groups, and radical series—all of which are key to understanding the exterior algebra structure linked to your algebra.

• SageMath
  Sage is a general-purpose math software that has solid built-in support for quivers and path algebras via its representation theory modules. While it’s not as specialized as QPA, it can still compute AR quivers for most common cases (especially over finite fields or rationals).

  For exterior algebras specifically, Sage has dedicated classes that let you define and manipulate these structures directly. You can combine this with its quiver tools to connect exterior algebra properties to the AR quiver of your path algebra. The syntax is pretty user-friendly: for example, define a quiver with Quiver([1,2], [(1,2,'a')]), build the path algebra, then call ar_quiver() on the algebra’s module category to get your AR quiver.

• Macaulay2
  Macaulay2 is primarily focused on commutative algebra and algebraic geometry, but it can handle non-commutative path algebras too, with support for computing AR quivers in relevant contexts. It’s especially useful if you’re interested in the overlap between exterior algebras and commutative algebra invariants—things like homology, free resolutions, and other structural details that tie into AR theory for certain algebra classes.

A quick pro tip: When working with exterior algebra structures, you’ll usually need to define the exterior algebra as a quotient of a path (or free) algebra by the appropriate ideal relations. All three tools above let you create quotient algebras, so you can set up your exterior algebra first, then compute its AR quiver or related module structures without hassle.

备注：内容来源于stack exchange，提问作者PlayerUnknown1098