mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly the study of

Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

, a Dunkl operator is a certain kind of

mathematical operator In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', bu ...

, involving

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

s but also

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

s in an underlying space. Formally, let ''G'' be a

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

with reduced root system ''R'' and ''k''_''v'' an arbitrary "multiplicity" function on ''R'' (so ''k''_''u'' = ''k''_''v'' whenever the reflections σ_''u'' and σ_''v'' corresponding to the roots ''u'' and ''v'' are conjugate in ''G''). Then, the Dunkl operator is defined by: :

T_i f(x) = \frac f(x) + \sum_ k_v \frac v_i

where

v_i

is the ''i''-th component of ''v'', 1 ≤ ''i'' ≤ ''N'', ''x'' in ''R''^''N'', and ''f'' a smooth function on ''R''^''N''. Dunkl operators were introduced by . One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy

T_i (T_j f(x)) = T_j (T_i f(x))

just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.

References

*{{Citation , last1=Dunkl , first1=Charles F. , title=Differential-difference operators associated to reflection groups , doi=10.2307/2001022 , mr=951883 , year=1989 , journal=

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...

, issn=0002-9947 , volume=311 , issue=1 , pages=167–183, doi-access=free Lie groups