Related papers: Extremal factorization lengths of elements in comm…

We construct equivalences among four notions associated to a reductive group scheme $G$: factorization super central extensions of the loop group of $G$ by $\mathbb G_m$ subject to a condition on the commutator, factorization super line…

Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. If an element $a \in H$ has a factorization $a=u_1 \cdot \ldots \cdot u_k$ into irreducible elements $u_1, \ldots, u_k \in H$, then…

In this paper we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of $M_{n}$, $GL_{n}$ and $SL_{n}$ are the centralizers of the trace $x_{1,1}+\dots+x_{n,n}$ in each algebra, for…

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a normal faithful state can be dilated by a group of Markov $*$-automorphisms analogous to the…

We study the properties of groups that have presentations in which the square of each generator gives the identity and all relations are of even length. We consider the parabolic subgroups of such a group and show that every element has a…

Let $M$ be a commutative cancellative monoid. The set $\Delta(M)$, which consists of all positive integers which are distances between consecutive factorization lengths of elements in $M$, is a widely studied object in the theory of…

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization…

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of…

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets…

Let $\mathcal P(S)$ be the semigroup obtained by equipping the family of all non-empty subsets of a (multiplicatively written) semigroup $S$ with the operation of setwise multiplication induced by $S$ itself. We call a subsemigroup $P$ of…

The aim of this article is to study the ideal class monoid $\mathcal{C}\ell(S)$ of a numerical semigroup $S$ introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $\mathcal{C}\ell(S)$. We observe that…

For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely…

A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal C_m…

Most factorization invariants in the literature extract extremal factorization behavior, such as the maximum and minimum factorization lengths. Invariants of intermediate size, such as the mean, median, and mode factorization lengths are…

For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factorizations of semigroup elements. This work generalizes many…

We determine subnormalisers of semisimple elements of prime power order in finite quasi-simple groups of Lie type. For this, we determine the maximal overgroups of normalisers of Sylow tori. This is motivated by the recent character…

A problem by Feichtinger, Heil, and Larson asks whether every infinite matrix $A$ with $\sum_{k,l}|A_{kl}| < \infty$ (an equivalent substitute for the Feichtinger algebra) that is positive-semidefinite admits a symmetric rank-one…

Given a join semilattice $S$ with a minimum $\hat{0}$, the quarks (also called atoms in order theory) are the elements that cover $\hat{0}$, and for each $x \in S \setminus \{\hat{0}\}$ a factorization (into quarks) of $x$ is a minimal set…

We study the decomposition matrices for the unipotent $\ell$-blocks of finite special unitary groups SU$_n(q)$ for unitary primes $\ell$ larger than $n$. Up to very few unknown entries, we give a complete solution for $n=2,\ldots,10$. We…

We consider the problem of summarizing a multi set of elements in $\{1, 2, \ldots , n\}$ under the constraint that no element appears more than $\ell$ times. The goal is then to answer \emph{rank} queries --- given $i\in\{1, 2, \ldots ,…