Related papers: Pop-Stack Operators for Torsion Classes and Cambri…
Each rooted plane tree $\mathsf{T}$ has an associated ornamentation lattice $\mathcal{O}(\mathsf{T})$. The ornamentation lattice of an $n$-element chain is the $n$-th Tamari lattice. We study the pop-stack operator…
Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$, we define the Coxeter pop-tsack torsing operator $\mathsf{Pop}_T:W\to W$ by $\mathsf{Pop}_T(w)=w\cdot\pi_T(w)^{-1}$, where $\pi_T(w)$ is the join in the…
Given a complex simple Lie algebra $\mathfrak g$ and a dominant weight $\lambda$, let $\mathcal B_\lambda$ be the crystal poset associated to the irreducible representation of $\mathfrak g$ with highest weight $\lambda$. In the first part…
For a finite irreducible Coxeter group $(W,S)$ with a fixed Coxeter element $c$ and set of reflections $T$, Defant and Williams define a pop-tsack torsing operation $\mathrm{Popt}\colon W \to W$ given by $\mathrm{Popt}(w) = w \cdot…
For each complete meet-semilattice $M$, we define an operator $\mathsf{Pop}_M:M\to M$ by \[\mathsf{Pop}_M(x)=\bigwedge(\{y\in M:y\lessdot x\}\cup\{x\}).\] When $M$ is the right weak order on a symmetric group, $\mathsf{Pop}_M$ is the…
Let $W$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $\mathsf{Pop}:W\to W$ to be the map that fixes the identity element and sends each nonidentity element $w$ to the meet of the elements covered by $w$…
Extending the classical pop-stack sorting map on the lattice given by the right weak order on $S_n$, Defant defined, for any lattice $M$, a map $\mathsf{Pop}_{M}: M \to M$ that sends an element $x\in M$ to the meet of $x$ and the elements…
Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator $\mathsf{Pop}_M : M \to M$ for each complete meet-semilattice $M$ by $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}).$$ This paper…
Given an essential semilattice congruence $\equiv$ on the left weak order of a Coxeter group $W$, we define the Coxeter stack-sorting operator ${\bf S}_\equiv:W\to W$ by ${\bf S}_\equiv(w)=w\left(\pi_\downarrow^\equiv(w)\right)^{-1}$, where…
We introduce a canonical operator-theoretic construction associated to a finite geometric lattice, in which a simple nonassociative ``diamond product'' on the lattice basis gives rise to a family of creation operators indexed by atoms and a…
An extended two-hadron operator is developed to extract the spectra of irreducible representations (irreps) in the finite volume. The irreps of the group for the finite volume system are projected using a coordinate-space operator. The…
Let $\Lambda$ be a finite-dimensional associative algebra. The torsion classes of $mod\, \Lambda$ form a lattice under containment, denoted by $tors\, \Lambda$. In this paper, we characterize the cover relations in $tors\, \Lambda$ by…
The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set $\operatorname{\mathsf{tors}} A$ of torsion classes over a finite-dimensional algebra $A$. We show that $\operatorname{\mathsf{tors}}…
We present a systematic method for constructing lattice QCD operators for systems of an arbitrary number of particles with arbitrary momentum, spin, and internal quantum numbers. Explicit constructions are provided for one-, two-, three-,…
The design and implementation of large sets of spatially-extended, gauge-invariant operators for use in determining the spectrum of baryons in lattice QCD computations are described. Group-theoretical projections onto the irreducible…
For ordered normed vector spaces $X, Y$, we consider the space $\mathcal{L}(X,Y)$ of bounded linear operators and characterize when its cone of positive operators has non-empty interior. When this is satisfied, we give a functional…
In one dimension, the exponential position operators introduced in a theory of polarization are identified with the twisting operators appearing in the Lieb-Schultz-Mattis argument, and their finite-size expectation values $z_L$ measure the…
The approximate representation of operators by finite matrices is analysed in terms of accuracy and convergence. The identity operator, for example, can be reconstructed using a basis of harmonic oscillator states leading to a narrow peak…
This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…
This paper first gives a necessary and sufficient condition that a lattice $L$ can be represented as the collection of all up-sets of a poset. Applying the condition, it obtains a necessary and sufficient condition that a lattice can be…