# A commutative monoid associated with a finite abelian group

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as $$v_{m,n} = e_m + e_n - e_{m+n}\,.$$ These elements generate the kernel of the surjective morphism $\mathbb{Z}^M/\langle e_0\rangle \longrightarrow M$ with $e_m \longrightarrow m$. Now define $K_+$ as the monoid generated by the $v_{m,n}$, for all $m,n \in M$. I am interested in the algebra $R[K_+]$, where $R$ is a field, or $R = \mathbb Z$.

This monoid plays an important role in my study of a special class of Galois abelian covers. For those who know this subject, $\mathop{\rm Spec}R[K_+]$ is an open subset of the main component of the toric (or Nakamura) Hilbert scheme $\mathop{\rm Hilb}^M (\mathbb{A}^{M\smallsetminus \{0\}})$.

Notice that, in general, $\mathop{\rm Spec}R[K_+]$ is not a toric variety, because it is not normal, since the monoid $K_+$ is not saturated. Its normalization is a toric variety.

Here are my questions.

Has this particular monoid, or its algebra, been studied before?

Can one describe the indecomposable elements and the extremal rays of the dual cone $K^\vee_{+}$ of $K_{+}$?

Set $K$ for the group generated by $K_+$ and $S = K^{\vee}_{+} \cap K^*$. For an indecomposable element we mean an element of $S$ which is not sum of non zero elements of $S$. The set of the indecomposable elements is the only minimal set of generators of $S$ and each extremal ray of $K^{\vee}_{+}$, opportunely normalized, gives an indecomposable element.

Up to scalars, an element $f$ of $S$ can be thought of as a collection of elements $a_m \in \mathbb{N}$, for $m \in M$, such that $a_m + a_n \geq a_{m+n}$ and $a_0 = 0$. $f$ is then given by $f(v_{m,n})=a_m+a_n-a_{m+n}$.

A special class of indecomposable elements arises in this way: suppose to have a basis of $K$ of the form $v_{m_1,n_1},\dots,v_{m_r,n_r}$, where $r = |M|-1$, and denote by $f_1,\dots,f_r$ its dual basis. If $f_1$ is non-negative over all the $v_{m,n}$, then $f_1$ is indecomposable. So a related question is determining what kinds of bases we can obtain in this way.

The only class of indecomposable elements I know is the following: if $\phi \colon M \longrightarrow \mathbb{Z}/l\mathbb{Z}$ is a surjective morphism take as $a_m$ the only integer between $0$ and $l-1$ such that $a_m \equiv \phi(m)$ mod $l$. In this way we obtain $|M|-1$ distinct elements, which are in general linearly dependent.

Any suggestion on how to handle this combinatorial problem will be very much appreciated!

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