First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally we provide an example that seems to contradict that claim. The question is

**Question**: Is the example given below a counterexample to the claim? And if not, why not?

### Definitions

Fix an Artinian monomial ideal $I$ of a polynomial ring $K[\mathbf{x}] = K[x_1, \dots, x_n]$. A monomial $\mathbf{x}^{\mathbf{b}}$ is a *socle monomial* of $I$ if $\mathbf{x}^{\mathbf{b}} \notin I$ and $x_i\mathbf{x}^{\mathbf{b}} \in I$ for all $i$. Let $\mathrm{MonSoc}(I)$ be the set of all socle monomials of $I$.

**Def**: $I$ is *reflection invariant* if there is a *canonical monomial* $\mathbf{x}^{\mathbf{K}}$ such that the map that sends a monomial $\mathbf{x}^{\mathbf{b}} \mapsto \mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}}$ is an involution on $\mathrm{MonSoc}(I)$.

Following these definitions the authors note the following.

### The Claim

**Claim**: $I$ is reflection invariant with canonical monomial $\mathbf{x}^{\mathbf{K}}$ if and only if the monomial ideal generated by $\mathrm{MonSoc}(I)$ equals the Alexander dual $I^{[\mathbf{K} + \mathbf{e}]}$ where $\mathbf{e} = (1,1,\dots, 1)$.

### The (Counter?) Example

Let $I = \langle a^4,~ab^2,~b^3,~a^3c,~abc,~c^3 \rangle \subset K[a,b,c]$ and let $\mathbf{K} = (3,2,2)$. Then

$$\mathrm{MonSoc}(I) = \left\{a^{3}b,~a^{2}c^{2},~b^{2}c^{2}\right\}.$$

By (the constructive proof of) Proposition 5.2 in this paper the ideal $J = \langle a^4, a^2b, b^3, ac, b^2c, c^3 \rangle$ is the unique Artinian ideal with

$$\mathrm{MonSoc}(J) = \left\{\mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}} \mid \mathbf{x}^{\mathbf{b}} \in \mathrm{MonSoc}(I)\right\}.$$

Moreover, the same algorithm can be used to show that $I$ is the unique Artinian ideal with

$$\mathrm{MonSoc}(I) = \left\{\mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{c}} \mid \mathbf{x}^{\mathbf{c}} \in \mathrm{MonSoc}(J)\right\}.$$

In particular, the map that sends a monomial $\mathbf{x}^{\mathbf{b}} \mapsto \mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}}$ is an involution on $\mathrm{MonSoc}(I)$, so $I$ is reflection-invariant with canonical monomial $\mathbf{x}^{\mathbf{K}}$. We now get a contradiction to the claim above by computing the Alexander dual (in Macaulay2, for example) and noting that the minimal generators of $I^{[(4,3,3)]}$ are $\{a^4bc,~a^2b^3c,~ab^2c^3\} \neq \mathrm{MonSoc}(I)$.

Again, the question is

**Question**: Is the example just given a counterexample to the above claim? And if not, why not?