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Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some…

In this paper, the finite size Dicke model of arbitrary number of qubits is solved analytically in an unified way within extended coherent states. For the $N=2k$ or $2k-1$ Dicke models ($k$ is an integer), the $G$-function, which is only an…

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the…

We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new…

In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid…

Coxeter groups are equipped with a partial order known as the weak order, such that $u \leq v$ if the inversions of $u$ are a subset of the inversions of $v$. In finite Coxeter groups, weak order is a complete lattice, but in infinite…

In this paper we show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of…

Fix an positive integer $n$. Let $K\subseteq\mathbb{R}^n$ be a compact set such that $K+\mathbb{Z}^n=\mathbb{R}^n$. We prove, via Algebraic Topology, that the integer points of the difference set of $K$, $(K-K)\cap\mathbb{Z}^n$, is not…

In this paper we present a method of constructing a nonlinear accelerator lattice that has an approximate integral of motion that is given upfront. The integral under consideration is a Hamiltonian in normalized (canonical) coordinates that…

A novel type of symmetry breaking is reported for the two-channel Kondo lattice where conduction electrons have spin and orbital (channel) degrees of freedom. Using the continuous-time quantum Monte Carlo and the dynamical mean-field…

Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison's anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands,…

The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been…

For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in…

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $kGe$ be a block of a group algebra of a finite group $G$, with normal defect group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a matrix…

Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation…

This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a\vee b in the section [b,1] is denoted by a\rightarrow b and can be considered as the connective…

The hyperfine interaction between the electron spin and the nuclear spins is one of the main sources of decoherence for spin qubits when the nuclear spins are disordered. An ordering of the latter largely suppresses this source of…

An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal…

Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…

Suppose that $W$ is a finite Coxeter group and $W_J$ a standard parabolic subgroup of $W$. The main result proved here is that for any for any $w \in W$ and reduced expression of $w$ there is an Elnitsky tiling of a $2m$-polygon, where $m =…