Multi-graded nilpotent tuples

TL;DR

This paper investigates the classification problem of multi-graded nilpotent tuples, generalizing graded nilpotent pairs, by analyzing their shapes and representation types through multi-staircase algebras.

Contribution

It introduces the concept of multi-graded nilpotent tuples, defines their shapes, and classifies their representation types using multi-staircase algebras.

Findings

Classification of representation types for multi-staircase algebras

Determination of finiteness conditions for multi-graded nilpotent tuples

Extension of nilpotent pair theory to multi-graded settings

Abstract

We discuss multi-graded nilpotent tuples of multi-graded vector spaces which are a generalization of graded nilpotent pairs. The multi-grading yields a natural notion of a shape of such tuple and our main interest is to answer the question "Is the number of multi-graded nilpotent tuples of a fixed shape, up to base change in the homogeneous components, finite?" Our methods make use of a translation to the class of so-called "Multi-staircase algebras" and we classify their representation types.