| 41. | Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group, or the group of rotations.
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| 42. | Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group, or the group of rotations.
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| 43. | The special orthogonal groups have additional spin representations that are not tensor representations, and are " typically " not spherical harmonics.
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| 44. | When " G " is a classical group, such as a symplectic group or orthogonal group, this is particularly transparent.
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| 45. | An example is the orthogonal group, defined by the relation M T M = I where M T is the transpose of M.
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| 46. | Reflectional spherical symmetry is isomorphic with the orthogonal group O ( 3 ) and has the 3-dimensional discrete point groups as subgroups.
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| 47. | These rotations form the special orthogonal group SO ( ), which can be represented by the group of orthogonal matrices with determinant 1.
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| 48. | If " K " does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1.
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| 49. | B r has an associated centerless compact groups the odd special orthogonal groups, SO ( 2 " r " + 1 ).
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| 50. | The matrix is a member of the three-dimensional special orthogonal group,, that is it is an orthogonal matrix with determinant 1.
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