CAxisTensor -- the signature tensor of the canonical axis path at a given level

Usage:

s = CAxisTensor(k, R)
Inputs:
- k, an integer, the level of the signature tensor to compute;
- R, an instance of the type NCPolynomialRing, the output tensor space;
Outputs:
- s, an instance of the type NCRingElement, an element of R; the k-th level signature of the canonical axis path in $\mathbb R^d$, where $d$ is the number of generators of R;

Description

The canonical axis path in $\mathbb{R}^d$ is the path from $(0, \dots, 0)$ to $(1, \dots, 1)$ given by $d$ linear steps in the unit directions $e_1, \dots, e_d$, in this order. The $k$-th level signature tensor of such a path has a combinatorial closed-form description (see the reference below) and can be obtained as follows:

i1 : d = 2;

i2 : k = 3;

i3 : R = wordAlgebra(d);

i4 : Cd = CAxisTensor(k, R); Cd // wordFormat -- k-th level signature of the canonical axis path in R^d

     1             1             1             1
o5 = - [2, 2, 2] + - [1, 2, 2] + - [1, 1, 2] + - [1, 1, 1]
     6             2             2             6

To expand on the example, we verify that the result agrees with the one obtained from sig. Notice that the matrix of increments for the canonical axis path in dimension $d$ is the $d \times d$ identity matrix.

i6 : M = id_(QQ^d); -- identity matrix

              2       2
o6 : Matrix QQ  <-- QQ

i7 : CAxisPath = pwLinPath(M) -- the canonical axis path in dimension d

o7 = Path in 2-dimensional space with 2 polynomial segments:

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

o7 : Path

i8 : Cd2 = sig(CAxisPath, k); Cd2 // wordFormat -- the k-th level signature

     1             1             1             1
o9 = - [2, 2, 2] + - [1, 2, 2] + - [1, 1, 2] + - [1, 1, 1]
     6             2             2             6

References

Varieties Of Signature Tensors (doi.org/10.1017/fms.2019.3)

Ways to use CAxisTensor:

CAxisTensor(ZZ,NCPolynomialRing)

For the programmer

The object CAxisTensor is a method function.

The source of this document is in PathSignatures/documentation.m2:1174:0.