Related papers: The cyclic sieving phenomenon: a survey

Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on…

For $c \in \mathbb{Q}$, consider the quadratic polynomial map $\varphi_c(x)=x^2-c$. Flynn, Poonen and Schaefer conjectured in 1997 that no rational cycle of $\varphi_c$ under iteration has length more than $3$. Here we discuss this…

Let $X(Q)=QC$ be a group, where $Q$ is a generalized quaternion group and $C$ is a cyclic group such that $Q\cap C=1$. In this paper, $X(Q)$ will be characterized and moreover, a complete classification for that will be given, provided $C$…

We present the proof of the cyclic sieving conjectures for generalised non-crossing partitions associated to well-generated complex reflection groups due to Armstrong, respectively to Bessis and Reiner, for the 26 exceptional well-generated…

We say that a cyclotomic polynomial Phi_{n}(x) has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Phi_{n}(x) and M(p) be the maximum of A(pqr). In 1968, Sister…

We consider the cyclotomic identity testing (CIT) problem: given a polynomial $f(x_1,\ldots,x_k)$, decide whether $f(\zeta_n^{e_1},\ldots,\zeta_n^{e_k})$ is zero, where $\zeta_n = e^{2\pi i/n}$ is a primitive complex $n$-th root of unity…

For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the…

A cyclic Riemannian Lie group is a Lie group $G$ equipped with a left-invariant Riemannian metric $h$ that satisfies $\oint_{X,Y,Z}h([X,Y],Z)=0$ for any left-invariant vector fields $X,Y,Z$. The initial concept and exploration of these Lie…

Applying E. Kowalski's recent generalization of the large sieve we prove that certain properties expected to be typical (irreducibility of the characteristic polynomial, absence of squares among the matrix coefficients...) are indeed…

We consider an example of four valued semantics partially inspired by quantum computations and negation-like operations occurred therein. In particular we consider a representation of so called square root of negation within this four…

Motivated by permutation statistics, we define for any complex reflection group W a family of bivariate generating functions. They are defined either in terms of Hilbert series for W-invariant polynomials when W acts diagonally on two sets…

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied…

Let $G$ be a group and let $\mathcal{F}$ be a family of subgroups of $G$ closed under conjugation. For a positive integer $n$, let $C_n$ denote a cyclic group of order $n$. We show that if there exists an integer $n$ such that every group…

In 1970, Coxeter gave a short and elegant geometric proof showing that if $p_1, p_2, \ldots, p_n$ are vertices of an $n$-gon $P$ in cyclic order, then $P$ is affinely regular if, and only if there is some $\lambda \geq 0$ such that…

The aim of this work is to extend the study of super-coinvariant polynomials, to the case of the generalized symmetric group $G_{n,m}$, defined as the wreath product $C_m\wr\S_n$ of the symmetric group by the cyclic group. We define a…

Recently, Armon and Swanson introduced signed standard tableaux and a corresponding super major index that refines the classical major index. In this paper, we prove that signed standard tableaux of rectangular shape exhibit a cyclic…

This paper deals with sequences of random variables $X_n$ only taking values in $\{0,\ldots,n\}$. The probability generating functions of such random variables are polynomials of degree $n$. Under the assumption that the roots of these…

In this paper, we compute explicitly the $q$-dimensions of highest weight crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving…

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime…

We prove a cyclic sieving result for the set of $3 \times k$ packed increasing tableaux with maximum entry $m :=3+k$ under K-promotion. The "curiosity" is that the sieving polynomial arises from the $q$-hook formula for standard tableaux of…