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# 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PathSignatures/documentation.m2:1174:0.