Co.Al.A.R.
Computational Algebraic Analysis Results
 

THIS PAGE IS DEVOTED TO SOME RESULTS IN CLIFFORD ANALYSIS and related topics


last update: March 2nd, 2006

for any question on this site or any of the CoAlA webpages, please contact adamiano@gmu.edu

==> Clebsch-Gordan coefficient calculation for the Rarita-Schwinger operator in dimension 3
 

        credit go to Y. Homma for the pdf file he has kindly sent us.  See also [YH]

==> Rarita-Schwinger complex for any number of operators in real dimension 3 [BD]: 
 
here H is the vector space of quaternions and it is naturally identified with the 4-dimensional representation space V3 of SU(2)
 
   ... --->  (H) b3  --->  (H) b2 --->  (H) b1 --->  (H) b0 --->  0
 

Number of operators
 b0
 b1
 b2
 b3
 b4
 b5
 b6 b7
n=1
 1
 1



 
n=2
 1
 2
 1


 
n=3
 1
 3
 10
 15
 9
 2
  
n=4
 1
 4
 28
 70
 84
 56
 20
 3

 

in general:         b0=1   ,   b1= n    ...    bj = n*(2n-1 choose j)*(j-1)/(j+1)

==> Our latest calculation of the Betti numbers for the resolution associated to RS+ and RS- , two operators naturally related to the Rarita-Schwinger operator because they come from a different invariant projection of S3 Ä (R3)* into irreducibles.
 
NOTE: here the Betti numbers have not been divided by four since the symbols of the operators are not square matrices.
            all number represent linear syzygies unless differently specified

 

RS+
 b0
 b1
 b2
 b3
 b4
 b5
 b6
 b7
 b8 b9
n=1
 4
 6 6 4
  


   
n=2
 4
 12 56 cubic + 8 linear 168 192 100 20
   
n=3
 4 18 252 cubic + 24 linear 1344 quadr + 10 linear
 3132
 4140 3550 1656 462 56
                     
RS-
 b0
 b1
 b2
 b3
 b4
 b5
 b6
 b7
 b8 b9
n=1
 4
 2      


   
n=2
 4
 4          
   
n=3
 4 6 2 quadratic              
                     

 

 
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[BD]  J. Bureš, A. Damiano. Algebraic analysis of the Rarita-Schwinger system in real dimension three, preprint 2006
 
[YH] Y. Homma, The Higher Spin Dirac Operators on 3-Dimensional Manifolds. Tokyo J. Math. 24 (2001), no. 2, 579-596.