Related papers: Crystals, semistandard tableaux and cyclic sieving…
Symmetry-breaking bifurcations, where a flow state with a certain symmetry undergoes a transition to state with a different symmetry, are ubiquitous in fluid mechanics. Much can be understood about the nature of these transitions from…
The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge's $q=-1$ phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a…
In this paper, we propose a general mechanism for the existence of quasicrystals in spatially extended systems (partial differential equations with Euclidean symmetry). We argue that the existence of quasicrystals with higher order…
It is shown that tiling in icosahedral quasicrystals can also be properly described by cyclic twinning at the unit cell level. The twinning operation is applied on the primitive prolate golden rhombohedra, which can be considered a result…
In 2015, Archdeacon introduced the notion of Heffter arrays and showed the connection between Heffter arrays and biembedding m-cycle and an n-cycle systems on a surface. In this paper we exploit this connection and prove that for every n >=…
An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be…
We identify a subalgebra \pH_n of the extended affine Hecke algebra \eH_n of type A. The subalgebra \pH_n is a \u-analogue of the monoid algebra of \S_n \ltimes \ZZ_{\geq 0}^n and inherits a canonical basis from that of \eH_n. We show that…
We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z}, \end{equation*} where $f$ is uniformly almost…
In the space $\mathbb{C}$ of the parameters $\lambda$ of the unicritical polynomials family $f(\lambda,z)=f_\lambda(z)=z^d+\lambda$ of degree $d>1$, we establish a quantitative equidistribution result towards the bifurcation current (indeed…
Our goal is to find classes of convolution semigroups on Lie groups $G$ that give rise to interesting processes in symmetric spaces $G/K$. The $K$-bi-invariant convolution semigroups are a well-studied example. An appealing direction for…
We are interested in the asymptotics of the number of standard Young tableaux $f^{\lambda/\mu}$ of a given skew shape $\lambda/\mu$. We mainly restrict ourselves to the case where both diagrams are balanced, but investigate all growth…
The Garsia--Haiman module is a bigraded $\mathfrak{S}_n$-module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes an $\mathfrak{S}_n$-set $X$ to a graded polynomial ring. The orbit harmonics can be…
Stone-$\breve{C}$ech compactifications derived from a discrete semigroup $S$ can be considered as the spectrum of the algebra $\mathcal{B}(S)$ or as a collection of ultrafilters on $S$. What is certain and indisputable is the fact that…
Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that - via results of Edjvet and Vdovina - are…
We show that every graph $G$ on $n$ vertices with $\delta(G) \geq (1/2+\varepsilon)n$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $\Omega(\log n)$. The proof is based on a recently introduced approach…
We consider phase transitions, in the form of spontaneous symmetry breaking (SSB) bifurcations of solitons, in dual-core couplers with fractional diffraction and cubic self-focusing acting in each core, characterized by Levy index $\alpha$.…
In QCD at high density, the color-octet quark-antiquark condensate $\langle\overline\psi\gamma_0(\lambda^A)_C (\lambda^A)_F\psi\rangle$ is generally nonzero and dynamically breaks the $\mathrm{SU}(3)_C\times…
We demonstrate that chiral symmetry breaking occurs in the confining regime of QCD-like theories with $N_c$ colors and $N_f$ flavors. Our proof is based on a novel strategy, called `downlifting', by which solutions of the 't Hooft anomaly…
Our recent works revisit the proof of chiral symmetry breaking in the confining phase of four-dimensional QCD-like theories, i.e. $SU(N_c)$ gauge theories with $N_f$ flavors of vectorlike quarks in the fundamental representation. The…
We will exhibit several instances of the cyclic sieving phenomenon involving statistics and involutions on the following combinatorial families of objects: permutations, set partitions, perfect matchings, D-permutations (and its…