mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras...

in the affine root system. The Macdonald polynomials are polynomials in n variables x=(x1,...,xn), where n is the rank of the affine root system. They...

ordering of the root poset. ADE classification Affine root system Coxeter–Dynkin diagram Coxeter group Coxeter matrix Dynkin diagram root datum Semisimple...

Macdonald identity for the affine root system of type A1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. Jacobi's...

also, α r + 1 {\displaystyle \alpha _{r+1}} denote the highest root. Then the affine Coxeter group is generated by the ordinary (linear) reflections...

orthogonal polynomials in several variables, depending on the choice of an affine root system. They include many other families of multivariable orthogonal polynomials...

Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald (1972). They include as special cases...

algebraically closed fields. One classifies such Lie algebras via their root system, which can be represented by a Dynkin diagram. One then classifies Dynkin...

generalizes the multiplicity of a root of a polynomial in the following way. The roots of a polynomial f are points on the affine line, which are the components...

affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie...

identity, and is the Macdonald identity for a certain non-reduced affine root system. It is related to Euler's pentagonal number theorem. ∏ n ≥ 1 ( 1 −...

finite dimension and Σ {\displaystyle \Sigma } an affine root system on V {\displaystyle V} . An affine Hecke algebra is a certain associative algebra that...

to arbitrary finite or affine root systems, with Dyson's original conjecture corresponding to the case of the An−1 root system and Andrews's conjecture...

polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (C∨ n, Cn), and in particular satisfy analogues of Macdonald's...

non-reduced affine root system of type (C∨ 1, C1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system. They are defined...

D_{n},E_{6},E_{7},E_{8},F_{4},G_{2}} ), while affine type indecomposable matrices classify the affine Lie algebras (say over some algebraically closed...

and are the Macdonald polynomials for the special case of the A1 affine root system (Macdonald 2003, p.156). Askey & Ismail (1983) and Gasper & Rahman...

Affine shape adaptation is a methodology for iteratively adapting the shape of the smoothing kernels in an affine group of smoothing kernels to the local...

verify the positive definiteness of a given matrix) root-finding algorithm Affine arithmetic Solving ODEs rigorously (This feature includes external tools...

and to systems of equations. The purpose of Newton's method is to find a root of a function. The idea is to start with an initial guess at a root, approximate...

can be extended to a non-degenerate form, and so allows a root system analysis of the affine Lie algebra. Kac, V.G. (1990). Infinite-dimensional Lie algebras...

In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated...

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in...

representations in Rn spaces, for any n, and can be used to visualize any affine coordinate system in a real n-space. Vertices of a hypercube have coordinates (x1...

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of...

under them, so affine subsets, convex cones, and convex sets are generalizations of vector subspaces: a vector subspace is also an affine subspace, a convex...

can be applied to real-world data. Affine holonomy groups are the groups arising as holonomies of torsion-free affine connections; those which are not Riemannian...

depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the...

same root system. The implication of the axiomatic nature of a root system and Serre's theorem is that one can enumerate all possible root systems; hence...

Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which...