Brackets -- Brackets, Grassmann-Cayley Algebra, and Projective Geometry

Fix integers $n \ge d \ge 1,$ and let $X = (x_{i j})$ be an $n\times d$ matrix of distinct variables in the polynomial ring $k [x_{i j}]$ over a fixed field $k.$ Think of each row of $X$ is a point in the projective space $\mathbb{P}^{d-1}$ of dimension $(d-1)$ over $k,$ so that $X$ as represents a configuration of $n$ points in this projective space. Many interesting geometric properties of this point configuration can be expressed in terms of the maximal minors of $X.$

For notational convenience, it is common to write these minors in bracket notation. A bracket is an expression $[\lambda_1 \lambda_2 \ldots \lambda_d]$ representing the minor of $X$ whose rows are given by indices $1\le \lambda_1 < \lambda_2 < \ldots < \lambda_d \le n.$

Formally, we may consider the map of polynomial rings $$\psi_{n,d} : k \left[ [\lambda_{i_1} \cdots \lambda_{i_d}] \mid 1 \le i_1 < \ldots < i_d \le n \right] \to k [X],$$ $$ [\lambda_{i_1} \cdots \lambda_{i_d}] \mapsto \det \begin{pmatrix} x_{i_1, 1} & \cdots & x_{i_1, d} \\ \vdots & & \vdots & \\ x_{i_d 1} & \cdots & x_{i_d d}\end{pmatrix}. $$

The classical bracket ring $B_{n,d}$ is the image of this map. This is the homogeneous coordinate ring of the Grassmannian of $(n-1)$-dimensional planes in $\mathbb{P}^{d-1}$ under its Plücker embedding.

We thank Thomas Yahl for helpful contributions, and the organizers of the 2023 Macaulay2 workshop in Minneapolis, IMA staff, and acknowledge support from the National Science Foundation grant DMS 2302476.

Sturmfels, Bernd. Algorithms in invariant theory. Springer Science & Business Media, 2008.