Editing Dunkl operator

{{Short description|Mathematical operator}}
In [[mathematics]], particularly the study of [[Lie groups]], a '''Dunkl operator''' is a certain kind of [[mathematical operator]], involving [[differential operator]]s but also [[Reflection (mathematics)|reflection]]s in an underlying space.

Formally, let ''G'' be a [[Coxeter group]] with reduced root system ''R'' and ''k''<sub>''v''</sub> an arbitrary "multiplicity" function on ''R'' (so ''k''<sub>''u''</sub> = ''k''<sub>''v''</sub> whenever the reflections σ<sub>''u''</sub> and σ<sub>''v''</sub> corresponding to the roots ''u'' and ''v'' are conjugate in ''G''). Then, the '''Dunkl operator''' is defined by:

:<math>T_i f(x) = \frac{\partial}{\partial x_i} f(x) + \sum_{v\in R_+} k_v \frac{f(x) - f(x \sigma_v)}{\left\langle x, v\right\rangle} v_i</math>

where <math>v_i </math> is the ''i''-th component of ''v'',  1 ≤ ''i'' ≤ ''N'', ''x'' in ''R''<sup>''N''</sup>, and ''f'' a smooth function on ''R''<sup>''N''</sup>.

Dunkl operators were introduced by {{harvs|txt|authorlink=Charles F. Dunkl|first=Charles |last=Dunkl|year=1989}}. One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy <math>T_i (T_j f(x)) = T_j (T_i f(x))</math> 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]] | issn=0002-9947 | volume=311 | issue=1 | pages=167–183| doi-access=free | jstor=2001022 }}

[[Category:Lie groups]]