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Search: All articles in the CJM digital archive with keyword reflection group

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1. CJM 2011 (vol 64 pp. 1359)

Nozaki, Hiroshi; Sawa, Masanori
Note on Cubature Formulae and Designs Obtained from Group Orbits
In 1960, Sobolev proved that for a finite reflection group $G$, a $G$-invariant cubature formula is of degree $t$ if and only if it is exact for all $G$-invariant polynomials of degree at most $t$. In this paper, we find some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and moreover gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007) which classifies tight Euclidean designs invariant under the Weyl group of type $B$ to other finite reflection groups.

Keywords:cubature formula, Euclidean design, radially symmetric integral, reflection group, Sobolev theorem
Categories:65D32, 05E99, 51M99

2. CJM 1998 (vol 50 pp. 193)

Xu, Yuan
Intertwining operator and $h$-harmonics associated with reflection groups
We study the intertwining operator and $h$-harmonics in Dunkl's theory on $h$-harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator $V$ on the harmonics, the main result provides a method to compute the action of the intertwining operator $V$ on polynomials and to construct an orthonormal basis for the space of $h$-harmonics.

Keywords:$h$-harmonics, intertwining operator, reflection group
Categories:33C50, 33C45

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