We explore the connection between atomicity in Prüfer domains and their corresponding class groups. We observe that a class group of infinite order is necessary for non-Noetherian almost Dedekind and Prüfer domains of finite character to be atomic. We construct a non-Noetherian almost Dedekind domain and exhibit a generating set for the ideal class semigroup.

The vertex algebra ${𝒲}_{1+\infty ,c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\ge 1$, it was conjectured in the physics literature that ${𝒲}_{1+\infty ,-n}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of ${𝒲}_{1+\infty ,-n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation of ${\text{GL}}_n$,...

Let $𝔽$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $𝔽$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,...,n_k\le p-1$ such that $V\cong {\oplus }_{i=1}^k{V}_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $S{L}_2\left(ℂ\right)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring...

We investigate the invariant rings of two classes of finite groups $G\le \mathrm{GL}(n,F_q)$ which are generated by a number of generalized transvections with an invariant subspace $H$ over a finite field $F_q$ in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with...