Related papers: Metacommutation as a Group Action on the Projectiv…

Conway and Smith introduced the operation of metacommutation for pairs of primes in the ring of Hurwitz integers in the quaternions. We study the permutation induced on the primes of norm p by a prime of norm q under metacommutation, where…

Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $\xi$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and…

For q > 2, Carlitz proved that the group of permutation polynomials (PPs) over F_q is generated by linear polynomials and x^{q-2}. Based on this result, this note points out a simple method for representing all PPs with full cycle over the…

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod…

In a quaternion order of class number one, an element can be factored in multiple ways depending on the order of the factorization of its reduced norm. The fact that multiplication is not commutative causes an element to induce a…

In this paper, we make use of the classification results of low-degree permutation rational functions together with their geometric properties to investigate rational functions that induce permutations on the multiplicative subgroup mu_q+1,…

Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general…

The Markoff group of transformations is a group $\Gamma$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong…

Let $p=3n+1$ be a prime with $n\in\mathbb{N}=\{0,1,\cdots\}$, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots<a_n<p$ be all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the sequence $$a_1\…

For prime levels $5 \le p \le 19$, sets of $\Gamma_{0}(p)$-permuted theta quotients are constructed that generate the graded rings of modular forms of positive integer weight for $\Gamma_{1}(p)$. An explicit formulation of the permutation…

Let $r$ be a positive integer, $h(X)\in\Bbb F_{q^2}[X]$, and $\mu_{q+1}$ be the subgroup of order $q+1$ of $\Bbb F_{q^2}^*$. It is well known that $X^rh(X^{q-1})$ permutes $\Bbb F_{q^2}$ if and only if $\text{gcd}(r,q-1)=1$ and…

Let $r\geq 3$ be any positive integer which is relatively prime to $p$ and $q^2\equiv 1 \pmod r$. Let $\tau_1, \tau_2$ be any permutation polynomials over $\mathbb{F}_{q^2},$ $\sigma_M$ is an invertible linear map over $\mathbb{F}_{q^2}$…

Let $p$ be a prime number. We consider diagonal $p$-permutation functors over a (commutative, unital) ring $\mathsf{R}$ in which all prime numbers different from $p$ are invertible. We first determine the finite groups $G$ for which the…

Let $H$ and $L$ be quantum groupoids. If $H$ has a quasitriangular structure, then we show that $L$ induces a Hopf algebra $C_{L}(L_s)$ in the category $_{H}\mathcal{M}$, which generalizes the transmutation theory introduced by Majid.…

The moduli space $\overline{\mathcal{M}}_{0,n}$ of $n$ pointed stable curves of genus $0$ admits an action of the symmetric group $S_n$ by permuting the marked points. We provide a closed formula for the character of the $S_n$-action on the…

Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup…

Let F* be the field of q elements and let P(n,q) denote the projective space of dimension n-1 over F*. We construct a family H^{n}_{k,i} of combinatorial homology modules associated to P(n,q) for a coefficient field F of positive…

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation…

The endomorphism algebras of the permutation modules for transitive permutation groups, known as Hecke algebras, are fundamental objects in representation theory. While group algebras are known to be symmetric over any field, it is natural…

The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura…