Related papers: The M\"{o}bius Function of the Suzuki Groups

This is basically a summary of [Mu]. The focus of the paper is the explicit computation of Hecke operators for period functions. In particular we compute the matrix representations of the 2nd Hecke operator on period functions for the full…

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{\"o}bius function on the lattice of…

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Suzuki curve $\mathcal{S}_q$. As the point $P$ varies, exactly two possibilities arise for $H(P)$: one for the…

We give an exact formula for the number of normal subgroups of each finite index in the Baumslag-Solitar group BS(p,q) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different…

The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group of type B, contains the generic Hecke algebra for the symmetric group, i.e. the Weyl group of type A, as a subalgebra. Inducing the index representation of the…

We introduce M\"obius functions of higher rank, a new class of arithmetic functions so that the classical M\"obius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free…

For $m \in {\mathbb N}$, let $S_m$ be the Suzuki curve defined over ${\mathbb F}_{2^{2m+1}}$. It is well-known that $S_m$ is supersingular, but the p-torsion group scheme of its Jacobian is not known. The a-number is an invariant of the…

The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an…

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We…

Let $(M, q)$ be a quadratic projective module of an odd rank over an commutative ring, where the form $q$ is semiregular, with global Witt index of at least $2$, and with $\mathrm{rk}(M) \ge 7$. We prove standard commutator formulae and…

We describe the subgroups of the group $\Z_m \times \Z_n \times \Z_r$ and derive a simple formula for the total number $s(m,n,r)$ of the subgroups, where $m,n,r$ are arbitrary positive integers. An asymptotic formula for the function…

We prove Sarnak's M\"obius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent…

We prove that the subgroup permutability degree of the simple Suzuki groups vanishes asymptotically. In the course of the proof we establish that the limit of the probability of a subgroup of $\Sz(q)$ being a 2-group is equal to 1.

We shed some light on the problem of determining the orbits of the braid group action on semiorthonormal bases of Mukai lattices as considered in \cite{GK04} and \cite{GO1}. We show that there is an algebraic (and in particular algorithmic)…

We present a class of reductions of M\"obius type for the lattice equations known as Q1, Q2, and Q3 from the ABS list. The deautonomised form of one particular reduction of Q3 is shown to exist on the $A_1^{(1)}$ surface which belongs to…

This article describes a practical approach for determining the lattice of subgroups U < V < G between given subgroups U and G, provided the total number of such subgroups is not too large. It builds on existing functionality for element…

We study the growth rate of the summatory function of the M\"obius function in the context of an algebraic curve over a finite field. Our work shows a strong resemblance to its number field counterpart, which was proved by Ng in 2004. We…

We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper…

We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\"{o}bius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and…

For a fixed odd prime $\ell$, we define a variant of the classical M\"{o}bius function on the poset of isomorphism classes of finite abelian $\ell$-groups, then we prove an analog of Hall's theorem on the vanishing of the M\"{o}bius…