spinRepresentationMatrices -- matrix generators for the spin representation of $\mathfrak{so}(2n)$

Usage:

spinRepresentationMatrices(n)
Inputs:
- n, an integer,
Optional inputs:
- CoefficientRing => ..., default value QQ
Outputs:
- L, a list,

Description

See [FH] Lecture 20.

In the example below, we compute matrix generators for the spin representation of $so_6$.

i1 : spinRepresentationMatrices(3)

o1 = {| -1/2 0   0   0    0   0    0    0   |, | -1/2 0   0    0   0    0  
      | 0    1/2 0   0    0   0    0    0   |  | 0    1/2 0    0   0    0  
      | 0    0   1/2 0    0   0    0    0   |  | 0    0   -1/2 0   0    0  
      | 0    0   0   -1/2 0   0    0    0   |  | 0    0   0    1/2 0    0  
      | 0    0   0   0    1/2 0    0    0   |  | 0    0   0    0   -1/2 0  
      | 0    0   0   0    0   -1/2 0    0   |  | 0    0   0    0   0    1/2
      | 0    0   0   0    0   0    -1/2 0   |  | 0    0   0    0   0    0  
      | 0    0   0   0    0   0    0    1/2 |  | 0    0   0    0   0    0  
     ------------------------------------------------------------------------
     0    0   |, | -1/2 0    0   0   0    0    0   0   |, | 0 0 0 0 0 0 0 0
     0    0   |  | 0    -1/2 0   0   0    0    0   0   |  | 0 0 0 0 0 0 0 0
     0    0   |  | 0    0    1/2 0   0    0    0   0   |  | 0 0 0 1 0 0 0 0
     0    0   |  | 0    0    0   1/2 0    0    0   0   |  | 0 0 0 0 0 0 0 0
     0    0   |  | 0    0    0   0   -1/2 0    0   0   |  | 0 0 0 0 0 1 0 0
     0    0   |  | 0    0    0   0   0    -1/2 0   0   |  | 0 0 0 0 0 0 0 0
     -1/2 0   |  | 0    0    0   0   0    0    1/2 0   |  | 0 0 0 0 0 0 0 0
     0    1/2 |  | 0    0    0   0   0    0    0   1/2 |  | 0 0 0 0 0 0 0 0
     ------------------------------------------------------------------------
     |, | 0 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 0 |, | 0 0 0 0  0 0 0 0 |, | 0 0
     |  | 0 0 1 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 -1 0 0 0 0 |  | 0 0
     |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0  0 0 0 0 |  | 1 0
     |  | 0 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 0 |  | 0 0 0 0  0 0 0 0 |  | 0 0
     |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0  0 0 1 0 |  | 0 0
     |  | 0 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0  0 0 0 0 |  | 0 0
     |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0  0 0 0 0 |  | 0 0
     |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 1 0 0 0 |  | 0 0 0 0  0 0 0 0 |  | 0 0
     ------------------------------------------------------------------------
     0 0 0 0  0 0 |, | 0 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0
     0 0 0 0  0 0 |  | 1 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0
     0 0 0 0  0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0
     0 0 0 0  0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 0 |  | 0 0 0 0 0 0
     0 0 0 0  0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0
     0 0 0 0  0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 1 0 0 0 |  | 0 0 0 0 0 0
     0 0 0 0  0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 1
     0 0 0 -1 0 0 |  | 0 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0
     ------------------------------------------------------------------------
     0 0 |, | 0 0 0 1 0 0 0 0 |, | 0 0  0 0 0 0 0 0 |, | 0 0 1 0 0 0 0 0  |,
     0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0  0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0  | 
     0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0  0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0  | 
     0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 -1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0  | 
     0 0 |  | 0 0 0 0 0 0 0 1 |  | 0 0  0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0  | 
     0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0  0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 -1 | 
     0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0  0 0 1 0 0 0 |  | 0 0 0 0 0 0 0 0  | 
     0 0 |  | 0 0 0 0 0 0 0 0 |  | 0 0  0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0  | 
     ------------------------------------------------------------------------
     | 0 1 0 0 0 0 0 0 |}
     | 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 1 |
     | 0 0 0 0 0 0 0 0 |

o1 : List

Ways to use spinRepresentationMatrices:

spinRepresentationMatrices(ZZ)

For the programmer

The object spinRepresentationMatrices is a method function with options.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2635:0.