CMonTensor -- the signature tensor of the canonical monomial 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 monomial path in $\mathbb R^d$, where $d$ is the number of generators of R;

Description

The canonical monomial path in $\mathbb{R}^d$ is the path from $(0, \dots, 0)$ to $(1, \dots, 1)$ given by $t\mapsto (t, t^2, \dots, t^d)$. Its $k$-th level signature has a 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 = CMonTensor(k, R); Cd // wordFormat --k-th level signature of the canonical monomial path in R^d

     1              1              2              1              4          
o5 = - [2, 2, 2] + -- [2, 2, 1] + -- [2, 1, 2] + -- [2, 1, 1] + -- [1, 2, 2]
     6             10             15             12             15          
     ------------------------------------------------------------------------
       1             1             1
     + - [1, 2, 1] + - [1, 1, 2] + - [1, 1, 1]
       6             4             6

To expand on the example, we verify that the result agrees with the one obtained from sig.

i6 : R=QQ[t];

i7 : CMonPath = polyPath(for i from 1 to d list t^i) -- The canonical axis path in dimension d

o7 = Path in 2-dimensional space with 1 polynomial segment:

           2
     {{t, t }}

o7 : Path

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

     1              1              2              1              4          
o9 = - [2, 2, 2] + -- [2, 2, 1] + -- [2, 1, 2] + -- [2, 1, 1] + -- [1, 2, 2]
     6             10             15             12             15          
     ------------------------------------------------------------------------
       1             1             1
     + - [1, 2, 1] + - [1, 1, 2] + - [1, 1, 1]
       6             4             6

References

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

Ways to use CMonTensor:

CMonTensor(ZZ,NCPolynomialRing)

For the programmer

The object CMonTensor is a method function.

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