GCExpression ^ GCExpression -- Shuffle product in the Grassmann-Cayley Algebra

Let $V$ be a vector space of dimension $d$ over a field $\mathbb{F}.$ We recall the exterior algebra, \[ \Lambda (V) = \displaystyle\bigoplus_{k=0}^d \Lambda^k (V), \] a set which forms an algebra with the usual addition and exterior product. The Grassmann-Cayley algebra is obtained by endowing this set with an additional shuffle product, defined below.

The linear span of $k$ independent vectors $a_1, \ldots , a_k\in V$ is represented by the extensor $a_1 \cdots a_k \in \Lambda^k (V)$, where the product is the usual exterior product. If we fix a basis $\{ e_1, \ldots , e_d \}$ for $V,$ then each exterior power $\Lambda^k (V)$ has a basis given by the extensors of the form $e_{i_1} \vee \cdots \vee e_{i_k}.$ We identify the extensor $a_1 \cdots a_d$ with the bracket $[a_1, \ldots , a_d]$ (see BracketRing.)

Let $A = a_1 a_2 \cdots a_j$ and $B = b_1 b_2 \cdots b_k$ be extensors with $j+k\ge d.$ The shuffle product of $A$ and $B$ represents the intersection of the subspaces represented by $A$ and $B.$ is defined by the formula \[ A \vee B = \displaystyle\sum_{\sigma \mid \text{is shuffle}} \operatorname{sgn}(\sigma ) [a_{\sigma (1)}, \ldots , a_{\sigma (d-k)}, b_1, \ldots , b_k], \] where the sum is taken over shuffle permutations with respect to the split $(d-k, j + k -d).$ These are the permutations $\sigma : [j] \to [j]$ satisfying $\sigma (1) < \sigma (2) < \ldots < \sigma (d-k)$ and $\sigma (d-k +1) < \sigma (d-k+2) < \cdots < \sigma (j).$ Extending $\mathbb{F}$-linearly, the shuffle product defines an associative, skew-commutative multiplication.

The exterior and shuffle products are implemented using the operators * and ^, respectively. When using Grassmann-Cayley algebras to prove theorems involving point configurations in $\mathbb{P} (V)$, it is beneficial to work with indeterminate points, rather than fixed elements of $V.$ Internally, we represent these indeterminate points with variables in a skew-commutative polynomial ring, $\mathbb{F}< a_1, \ldots , a_n>$ (see SkewCommutative.)

In the example below, three lines spanned by six distinct points in $\mathbb{P}^2$ in the Grassmann-Cayley algebra, as well as the intersection of these three lines. In the latter case, we obtain an element of the bracket ring $B_{3,6}$ (see BracketRing.)

i1 : G = gc(a..f, 3)

o1 = Grassmann-Cayley Algebra generated by 1-extensors a..f

o1 : GCAlgebra

i2 : lab = (a*b)_G

o2 = a*b

o2 : GCExpression

i3 : lcd = (c*d)_G;

i4 : lef = (e*f)_G;

i5 : lab ^ lcd ^ lef

o5 = -2*[bcd]*[aef]+2*[bef]*[acd]

o5 : GCExpression

See also GCAlgebra, GCExpression.