Completely reducible factors of harmonic polynomials of three variables

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

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HYPER-SPHERICAL HARMONICS AND ANHARMONICS IN m-DIMENSIONAL SPACE

International Journal of Modern Physics E ◽

10.1142/s0218301308010398 ◽

2008 ◽

Vol 17 (06) ◽

pp. 1125-1130

Author(s):

M. R. SHOJAEI ◽

A. A. RAJABI ◽

H. HASANABADI

Keyword(s):

Quantum Mechanics ◽

General Theory ◽

Spherical Harmonics ◽

Interaction Potential ◽

Dimensional Space ◽

Harmonic Polynomials ◽

Central Importance ◽

Computational Tools ◽

Relative Coordinate

In quantum mechanics the hyper-spherical method is one of the most well-established and successful computational tools. The general theory of harmonic polynomials and hyper-spherical harmonics is of central importance in this paper. The interaction potential V is assumed to depend on the hyper-radius ρ only where ρ is the function of the Jacobi relative coordinate x1, x2,…, xn which are functions of the particles' relative positions.

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COMPLETELY REDUCIBLE SETS

International Journal of Algebra and Computation ◽

10.1142/s0218196713400158 ◽

2013 ◽

Vol 23 (04) ◽

pp. 915-941 ◽

Cited By ~ 4

Author(s):

DOMINIQUE PERRIN

Keyword(s):

Closure Properties ◽

Linear Representations ◽

Complete Reducibility ◽

The Family ◽

Syntactic Representation ◽

Completely Reducible ◽

Rational Sets ◽

Cyclic Sets

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

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