halfspinRepresentationMatrices -- matrix generators for the halfspin representations of $\mathfrak{so}(2n)$

Usage:

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

Description

See [FH] Lecture 20. The parity of the second input p determines which of the two half-spin representations is returned. For $S^{+}$, enter an even integer. For $S^{-}$, enter an odd integer.

In the example below, we compute matrix generators for the spin representation of $so_6$. Then we compute its half-spin representations. In the bases used by the package, this decomposes the spin representation into the upper left and lower right blocks.

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

i2 : halfspinRepresentationMatrices(3,0)

o2 = {| 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 1 0  0 |  | 0  0 0 0 |  | 0 0 0 0 |  | 0 0 1 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 -1 |  | 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 0 0 0
     0 0 0 |  | 0 0 0 -1 |  | 0 0 0 0 |  | 1 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 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 1 |, | 0 0  0 0 |, | 0 0 1 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 0 |

o2 : List

i3 : halfspinRepresentationMatrices(3,1)

o3 = {| 1 0  0 0 |, | 0 0 0  0 |, | -1 0 0 0 |, | 0 1 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 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 0  0 |  | 0  0 0 1 |  | 0 0 0 0 |  | 0 0 0 0 |  | 1
     ------------------------------------------------------------------------
     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 1 0 0
     0 0 0 |  | 0 0 0 0 |  | 0 -1 0 0 |  | 0 0 1 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 |  | 1 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 |

o3 : List

Ways to use halfspinRepresentationMatrices:

halfspinRepresentationMatrices(ZZ,ZZ)

For the programmer

The object halfspinRepresentationMatrices 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.m2:2716:0.