sgnVolTensor -- The signed volume tensor in a free associative algebra.

Description

Given a free associative algebra $K \langle \mathtt{1}, \dots, \mathtt{d} \rangle$ over a field $K$, the signed volume tensor is $$\frac{1}{d!}\sum_{\sigma} (-1)^{\operatorname{sign}(\sigma)} \mathtt{\sigma(1)}\dots \mathtt{\sigma(d)}$$ where the sum is taken over all permutations of ${1, \dots, d}$. Under the isomorphism $\mathbb R \langle \mathtt{1}, \dots, \mathtt{d} \rangle \to T(\mathbb{R}^d), \ \mathtt{i} \mapsto e_i^*$ it corresponds to the determinant $e_1^* \wedge \dots \wedge e_d^*$.

This method computes the signed volume tensor of the given NCPolynomialRing. The output is in the same ring.

i1 : d = 3;

i2 : R = wordAlgebra(d)

o2 = R

o2 : NCPolynomialRing

i3 : sgnVolTensor(R) // wordFormat

       1             1             1             1             1            
o3 = - - [3, 2, 1] + - [3, 1, 2] + - [2, 3, 1] - - [2, 1, 3] - - [1, 3, 2] +
       6             6             6             6             6            
     ------------------------------------------------------------------------
     1
     - [1, 2, 3]
     6

The paper [1] explores under what conditions the value of the signature at the signed volume tensor computes the volume of the convex hull of a path. One instance where this is true is the case of canonical axis paths (see CAxisTensor). For example, for $\mathtt{d}=3$ the convex hull of the canonical axis path in $\mathbb{R}^{\mathtt{d}}$ is a tetrahedron, whose volume is $\frac{1}{6}$. We verify this.

i4 : X = linPath({1,0,0})**linPath({0,1,0})**linPath({0,0,1})

o4 = Path in 3-dimensional space with 3 polynomial segments:

     {{t, 0, 0}, {0, t, 0}, {0, 0, t}}

o4 : Path

i5 : R = wordAlgebra(3) -- where the signature of X lives

o5 = R

o5 : NCPolynomialRing

i6 : v = sgnVolTensor(R);

i7 : sig(X, v) -- computes the signature of X at the signed volume tensor

     1
o7 = -
     6

o7 : QQ

References

[1] Convex Hulls of Curves: Volumes and Signatures (doi.org/10.1007/978-3-031-38271-0_45)

Ways to use sgnVolTensor:

sgnVolTensor(NCPolynomialRing)

For the programmer

The object sgnVolTensor is a method function.

The source of this document is in PathSignatures/documentation.m2:918:0.