halfshuffle -- compute the half-shuffle of an ordered pair of words

Usage:

h = f >> g
Inputs:
- f, an instance of the type NCRingElement,
- g, an instance of the type NCRingElement,
Outputs:
- h, an instance of the type NCRingElement, the half-shuffle f>>g of the two words;

Description

We start with the mathematical definition, based on [1] (where the operation is called right half-shuffle). Let $T^{\geq 1}(\mathbb{R}^d)$ be the vector space spanned by the non empty words on $d$ letters. Then the half shuffle $>>$ is defined recursively to be $$ w >> i := wi$$ for $w$ a word and $i$ a letter and $$ w >> vi := (w >> v + v >> w)\bullet i$$ for $w, v$ words and $i$ a letter, where $\bullet$ is the concatenation product on words.

As stated in the reference, the shuffle on non empty words can be seen as a symmetrization of the half-shuffle. As a usage example, we verify this in a particular instance.

i1 : R = wordAlgebra(3);

i2 : w = [1]_R

o2 = Lt
       1

o2 : R

i3 : v = [1,2,3]_R

o3 = Lt Lt Lt
       1  2  3

o3 : R

i4 : s = w ** v --shuffle product of w, v

                                   2
o4 = Lt Lt Lt Lt +Lt Lt Lt Lt +2Lt  Lt Lt
       1  2  3  1   1  2  1  3    1   2  3

o4 : R

i5 : hsSymm = (w >> v) + (v >> w)--half-shuffle product symmetrization of w, v

                                   2
o5 = Lt Lt Lt Lt +Lt Lt Lt Lt +2Lt  Lt Lt
       1  2  3  1   1  2  1  3    1   2  3

o5 : R

i6 : s == hsSymm

o6 = true

References

[1] Signatures of paths transformed by polynomial maps (doi.org/10.1007/s13366-020-00493-9)

Ways to use halfshuffle:

halfshuffle(NCRingElement,NCRingElement)

For the programmer

The object halfshuffle is a method function.

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

halfshuffle -- compute the half-shuffle of an ordered pair of words

Description

References

See also

Ways to use halfshuffle:

For the programmer