- Source: Dunkl operator
In mathematics, particularly the study of Lie groups, a Dunkl operator is a certain kind of mathematical operator, involving differential operators but also reflections in an underlying space.
Formally, let G be a Coxeter group with reduced root system R and kv an arbitrary "multiplicity" function on R (so ku = kv whenever the reflections σu and σv corresponding to the roots u and v are conjugate in G). Then, the Dunkl operator is defined by:
T
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=
∂
∂
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+
∑
v
∈
R
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k
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−
f
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⟨
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{\displaystyle 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}}
where
v
i
{\displaystyle v_{i}}
is the i-th component of v, 1 ≤ i ≤ N, x in RN, and f a smooth function on RN.
Dunkl operators were introduced by Charles Dunkl (1989). One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy
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T
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=
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{\displaystyle 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
Dunkl, Charles F. (1989), "Differential-difference operators associated to reflection groups", Transactions of the American Mathematical Society, 311 (1): 167–183, doi:10.2307/2001022, ISSN 0002-9947, MR 0951883
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