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# 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PathSignatures/documentation.m2:918:0.