Function `spraycross()` function in the `spray` package
In spray: Sparse Arrays and Multivariate Polynomials

set.seed(0)
library("spray")
options(rmarkdown.html_vignette.check_title = FALSE)
knitr::opts_chunk$set(echo = TRUE)
knit_print.function <- function(x, ...){dput(x)}
registerS3method(
  "knit_print", "function", knit_print.function,
  envir = asNamespace("knitr")
)

knitr::include_graphics(system.file("help/figures/spray.png", package = "spray"))

spraycross
spraycross2

To cite the spray package in publications, please use @hankin2022_spray. Function spraycross() returns the tensor cross product of any number of spray objects (interpreted as tensors); spraycross2() is a helper function that returns the product of two such spray objects.

The tensor cross product

In a memorable passage, @spivak1965 states:

Integration on chains

If $V$ is a vector space over $\mathbb{R}$, we denote the $k$-fold product $V\times\cdots\times V$ by $V^k$. A function $T\colon V^k\longrightarrow\mathbb{R}$ is called *multilinear* if for each $i$ with $1\leqslant i\leqslant k$ we have $$ T\left(v_1,\ldots, v_i + {v'}_i,\ldots, v_k\right)= T\left(v_1,\ldots,v_i,\ldots,v_k\right)+ T\left(v_1,\ldots,{v'}_i,\ldots,v_k\right),\\ T\left(v_1,\ldots,av_i,\ldots,v_k\right)=aT\left(v_1,\ldots,v_i,\ldots,v_k\right) $$ A multilinear function $T\colon v^k\longrightarrow\mathbb{R}$ is called a *$k$-tensor* on $V$ and the set of all $k$-tensors, denoted by $\mathcal{J}^k(V)$, becomes a vector space (over $\mathbb{R}$) if for $S,T\in\mathcal{J}^k(V)$ and $a\in\mathbb{R}$ we define $$ (S+T)(v_1,\ldots,v_k) = S(v_1,\ldots,v_k) + T(v_1,\ldots,v_k) (aS)(v_1,\ldots,v_k) = a\cdot S(v_1,\ldots,v_k) $$ There is also an operation connecting the various spaces $\mathcal{J}(V)$. If $S\in\mathcal{J}^k(V)$ and $T\in\mathcal{J}^l(V)$, we define the *tensor product* $S\otimes T\in\mathcal{J}^{k+l}(V)$ by $$ S\otimes T(v_1,\ldots,v_k,v_{k+1},\ldots,v_{k+l})= S(v_1,\ldots,v_k)\cdot T(v_{k+1},\ldots,v_{k+l}). $$

- Michael Spivak, 1969 (Calculus on Manifolds, Perseus books). Page 75

Spivak goes on to observe that the tensor product is distributive and associative but not commutative. He then proves that the set of all $k$-fold tensor products

$$ \phi_{i_1}\otimes\cdots\otimes\phi_{i_k}\qquad 1\leqslant i_1,\ldots,i_k\leqslant n $$

[where $\phi_i(v_j)=\delta_{ij}$,$v_1,\ldots,v_k$ being a basis for $V$] is a basis for $\mathcal{J}^k(V)$, which therefore has dimension $n^k$. Function spraycross2() evaluates the tensor product and I give examples here.