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 = {| 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 1 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 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 -1 0 0 0 |  |
      | 0 0 0 0  0 -1 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 -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 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 -1
     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 |  | 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 |  | 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 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 |  | 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 1 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 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 -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
     |  | 0 0 0 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 -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
     ------------------------------------------------------------------------
     1 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 -1 |  | 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 |

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 /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:2680:0.