Dear Forum members,
I have a question on modules defined over polynomial rings. Let us define a
polynomial ring $R$ as
$F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$
where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field.
Let $G$ be a free $R$-module with rank $t$ and $P$
be a free $R$-module of Pauli operators. $\sigma$ is a map from
$G$ to $P$. Please advise how to write a code in GAP to calculate the
cokernel of such a map.
Just for example (taken from page 54 of arxiv.1305.6973 or page 41 of
arxiv.1607.01387),
though this is not essential for the question, I can have two
``interaction''
terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity
operator $I$ on 4 sites with two 2-dimensional systems per site as
$
II(0,0)-IX(0,1)-XI(1,0)-XX(1,1)
$
and
$
ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1)
$
where on each site $\left(x,y\right)$ (mentioned in the bracket after
the Pauli operators), the first(second) Pauli acts on the first(second)
two dimensional system on that site. The map $\sigma$ can be written
as
$
\sigma=\left(\begin{array}{cc}
y+xy & 0 \\
x+xy & 0 \\
0 & 1+y \\
0 & 1+x
\end{array}\right)
$
where for example, $y+xy$ is a polynomial that specifies the action
on the first two dimensional system as
$
y+x y=0 \hspace{1mm} x^0 y^0+ 1 \hspace{1mm} x^0 y^1+0 \hspace{1mm} x^1
y^0 +1 \hspace{1mm} x^1 y^1
$
where the exponents are the coordinates of the sites and coefficients
$0$ and $1$ imply whether there is a Pauli acting or not.
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