mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, particularly the study of

Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas ...

, a Dunkl operator is a certain kind of

mathematical operator Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include numb ...

, 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 retur ...

s but also reflections 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 ref ...

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 pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...

, issn=0002-9947 , volume=311 , issue=1 , pages=167–183, doi-access=free , jstor=2001022 Lie groups