Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that $\mathcal P_{\text{fin},0}(\mathbb N)$ has a unique non-trivial automorphism, and conjectured that the automorphism group of $\mathcal P_{\text{fin},0}(H)$ is trivial whenever $H \ne \mathbb N$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali--Yan theorem.
Let $\widetilde{S}\to S$ be an unbranched regular $p$-fold cyclic cover of a closed orientable surface $S$ of genus $g$. Two natural groups are associated with this cover. The first is the centralizer in $\mathrm{Mod}(\widetilde{S})$ of a chosen generator $\sigma$ of the deck transformation group, denoted by $\mathrm{M...
Let $X = \mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ be the thrice-punctured over a ring of $S$-integers $\mathcal{O}_{K,S}$ in a number field~$K$. For any quotient $\pi_1^{\mathrm{mot}}(X,0) \twoheadrightarrow \Pi$ of Deligne--Goncharov's motivic fundamental group there is an associated Selmer scheme which parametrise...
Let $\mathsf k$ be a field, $S=\mathsf k[x,y,z]$, and $R=S/I$ be a standard graded Artinian Gorenstein $\mathsf k$-algebra of codimension three. The $h$-vector of such an algebra is known to be symmetric and unimodal. Mir\'o-Roig proved that if $\mathsf k$ is algebraically closed of characteristic zero and the $h$-vect...
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the class...
Let $H_\Gamma$ be the Bestvina-Brady group associated to a finite connected graph $\Gamma$. For a biconnected defining graph, we prove two structure theorems. First, restriction induces an isomorphism $\mathrm{IAut}(A_\Gamma)\cong \mathrm{IAut}(H_\Gamma)$ compatible with the Andreadakis-Johnson filtrations. Second, the...
Let $n\geq5$ and $m\geq3$. Let $\Phi\colon\mathrm{B}_n\to\mathrm{B}_m$ be a homomorphism of braid groups. We prove that if the image of $\Phi$ is irreducible and not cyclic, then $m=n$ and $\Phi$ agrees with an automorphism modulo the center $Z(\mathrm{B}_m)$. This resolves in the affirmative a conjecture of Chen, Kord...
Rodrigo de Pool, Peter Huxford, Daniel Minahan et al.· 0 citations
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