An object of class BracketRing represents the bracket ring $B_{n,d}$. For example, let $n=4, d=2,$ so that $$X=\begin{pmatrix} x_{1,1}&x_{1,2}\\ x_{2,1}&x_{2,2}\\ x_{3,1}&x_{3,2}\\ x_{4,1}&x_{4,2}\\ \end{pmatrix}.$$ There are $6=\binom{4}{2}$ brackets, and the matrix $X$ represents a configuration of $6$ points on the projective line $\mathbb{P}^1.$ These brackets are not algebraically independent, as they satisfy the quadratic Plücker relation, $$ [1 2] [3 4] - [1 3] [2 4] + [1 4] [2 3] = 0. $$ Some basic syntax for working with objects of class BracketRing is illustrated in the documentation page bracketRing.
The object BracketRing is a type, with ancestor classes AbstractGCRing (missing documentation) < HashTable < Thing.
The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Brackets.m2:417:0.