Cocycle
Definition
Let \(S\) be a numerical semigroup and \(n \in S \setminus \{0\}\). If \(w(i)\) denotes the smallest element in \(S\) such that \(w(i) \equiv i ~~ (mod ~ n)\) for all \(i \in \{0, \ldots, n-1\}\) and \(\overline{m} = m ~ (mod ~ n)\) for all \(m \in \mathbb{N}\), it is defined the cocycle of \(S\) with respect to \(n\), denoted by \(H(S,n)\), as the matrix \(H(S,n) = (h_{i,j})_{i, j = 0}^{n-1}\), where \(h_{i,j}\) is defined as
\[ h_{i,j} = \frac{w(i) + w(j) - w\left(\overline{i+j}\right)}{n}. \]
It can be proven that \(h_{i,j} \in \mathbb{N}\) for all \(i,j \in \{0, \ldots, n-1\}\).
Examples
\(\circ\) Let \(S = \langle 5, 9, 11, 12 \rangle = \{0, 5, 9, 10, 11, 12, 14, \rightarrow\}\) and \(n = 5\). It is well known that \(Ap(S,n) = \{w(0) = 0, w(1), \ldots, w(n-1)\}\), where \(Ap(S,n)\) denotes the Apéry list of \(S\) with respect to \(n\). Since \(Ap(S,5) = \{0, 9, 11, 12, 18\}\), it is deduced that
\[ w(0) = 0, ~ w(1) = 11, ~ w(2) = 12, ~w(3) = 18, ~w(4) = 9, \]
and the cocycle of \(S\) with respect to \(n = 5\) is
\[ H(S,5) = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 4 & 4 \\ 0 & 1 & 3 & 6 & 2 \\ 0 & 4 & 6 & 5 & 3 \\ 0 & 4 & 2 & 3 & 0 \\ \end{pmatrix}. \]
The symmetry of the cocycle comes from the definition.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 6, 15, 17, 22 \rangle\), in GAP:
gap> S := NumericalSemigroup(6, 15, 17, 22);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an element \(n \in S \setminus \{0\}\), the function CocycleOfNumericalSemigroupWRTElement returns the cocycle of \(S\) with respect to \(n\).
gap> CocycleOfNumericalSemigroupWRTElement(S, 6);
[ [ 0, 0, 0, 0, 0, 0 ], [ 0, 7, 9, 5, 7, 9 ],
[ 0, 9, 7, 5, 9, 2 ], [ 0, 5, 5, 5, 0, 0 ],
[ 0, 7, 9, 0, 2, 4 ], [ 0, 9, 2, 0, 4, 2 ] ]Therefore, the cocycle of \(S\) with respect to \(n = 6\) is
\[ H(S, 6) = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 7 & 9 & 5 & 7 & 9 \\ 0 & 9 & 7 & 5 & 9 & 2 \\ 0 & 5 & 5 & 5 & 0 & 0 \\ 0 & 7 & 9 & 0 & 2 & 4 \\ 0 & 9 & 2 & 0 & 4 & 2 \\ \end{pmatrix}. \]
References
https://gap-packages.github.io/ numericalsgps.