scholarly journals Weight-lattice discretization of Weyl-orbit functions

2016 ◽  
Vol 57 (8) ◽  
pp. 083512 ◽  
Author(s):  
Jiří Hrivnák ◽  
Mark A. Walton
Keyword(s):  
Weight Lattice

scholarly journals Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 354 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Jiří Patera
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Bell Polynomials ◽  
The Third ◽  
Weight Lattice ◽  
Hybrid Character ◽  
Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

Colourings of quasicrystals

1994 ◽  
Vol 72 (7-8) ◽  
pp. 442-452 ◽  
Author(s):  
R. V. Moody ◽  
J. Patera
Keyword(s):  
Number Field ◽  
Penrose Tiling ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Arithmetic Properties ◽  
Weight Lattice

We introduce a notion of colouring the points of a quasicrystal analogous to the idea of colouring or grading of the points of a lattice. Our results apply to quasicrystals that can be coordinatized by the ring R of integers of the quadratic number field [Formula: see text] and provide a useful and wide ranging tool for determining of sub-quasicrystals of quasicrystals. Using the arithmetic properties of R we determine all possible finite colourings. As examples we discuss the 4-colours of vertices of a Penrose tiling arising as a subset of 5-colouring of an R lattice, and the 4-colouring of quasicrystals arising from the D6 weight lattice.

scholarly journals Lefschetz Operators, Hodge–Riemann Forms, and Representations

2020 ◽  
Author(s):  
Peter Fiebig
Keyword(s):  
Root System ◽  
Bilinear Form ◽  
Chevalley Group ◽  
Simple Root ◽  
Complex Geometry ◽  
Linear Operators ◽  
Vector Spaces ◽  
Tilting Module ◽  
Weight Lattice ◽  
Simply Connected

Abstract For a field of characteristic $\ne 2$, we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

scholarly journals Identification of Galectin-2–Mucin Interaction and Possible Formation of a High Molecular Weight Lattice

2017 ◽  
Vol 40 (10) ◽  
pp. 1789-1795 ◽  
Author(s):  
Mayumi Tamura ◽  
Dai Sato ◽  
Moeko Nakajima ◽  
Masanori Saito ◽  
Takaharu Sasaki ◽  
...  
Keyword(s):  
Molecular Weight ◽  
High Molecular Weight ◽  
Weight Lattice

Hamiltonian reduction on the weight lattice

1992 ◽  
Vol 381 (1-2) ◽  
pp. 431-447
Author(s):  
Karyn M. Apfeldorf
Keyword(s):  
Hamiltonian Reduction ◽  
Weight Lattice

scholarly journals On the Combinatorics of Row and Corner Transfer Matrices of the $A^{(1)}_{n-1}$ Restricted Face Models

1997 ◽  
Vol 12 (20) ◽  
pp. 3551-3586 ◽  
Author(s):  
Srinandan Dasmahapatra
Keyword(s):  
Spectral Data ◽  
Polynomial Identity ◽  
Spin Chain ◽  
Two Dimensional ◽  
Transfer Matrices ◽  
Dimensional Model ◽  
Corner Transfer Matrices ◽  
Weight Lattice ◽  
Index Sets ◽  
Two Dimensional Model

We establish a weight-preserving bijection between the index sets of the spectral data of row-to-row and corner transfer matrices for [Formula: see text] restricted interaction round a face (IRF) models. The evaluation of momenta by adding Takahashi integers in the spin chain language is shown to directly correspond to the computation of the energy of a path on the weight lattice in the two-dimensional model. As a consequence we derive fermionic forms of polynomial analogs of branching functions for the cosets [Formula: see text], and establish a bosonic–fermionic polynomial identity.

Crystals and Gelfand-Tsetlin Patterns

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Crystal Bases ◽  
Cartan Type ◽  
Weight Lattice ◽  
Crystal Graphs ◽  
Multiple Dirichlet Series

This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. The discussion is restricted to crystals of Cartan type Aᵣ. The Weyl vector, denoted by ρ‎, is considered as an element of the weight lattice, and the bijection between Gelfand-Tsetlin patterns and tableaux is described. The chapter also examines the λ‎-part of the multiple Dirichlet series in terms of crystal graphs.

scholarly journals Central Splitting of A2 Discrete Fourier–Weyl Transforms

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1828 ◽  
Author(s):  
Jiří Hrivnák ◽  
Mariia Myronova ◽  
Jiří Patera
Keyword(s):  
Reflection Group ◽  
Unitary Matrix ◽  
Weyl Transform ◽  
Matrix Decompositions ◽  
Weight Lattice ◽  
Affine Weyl Group ◽  
Weyl Transforms ◽  
Original Weight ◽  
Unitary Transform ◽  
Point Set

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

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