exteriorPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\bigwedge^k V$ for a $\mathfrak{g}$-module $V$

Function: exteriorPower
Usage:

exteriorPower(k,V)
Inputs:
- k, an integer,
- V, an instance of the type LieAlgebraRepresentation,
Optional inputs:
- Strategy => ..., default value null
Outputs:
- W, an instance of the type LieAlgebraRepresentation,

Description

Let $V$ be a LieAlgebraRepresentation. Then this function computes the action of $\mathfrak{g}$ on $W = \bigwedge^k V$.

In the example below, we compute $\bigwedge^2 V$ for the standard representation of $sl_3$.

i1 : V = standardRepresentation("A",2);

i2 : W = exteriorPower(2,V)

o2 = LieAlgebraRepresentation{"Basis" => Enhanced basis of 𝔞                                                                                          }
                                                            2
                              "Module" => LL   (𝔞 )
                                            0,1  2
                              "RepresentationMatrices" => {| 0 0 0  |, | 1 0  0 |, | 0 0 0 |, | 0 1 0 |, | 0 0 -1 |, | 0 0 0 |, | 0 0 0 |, | 0  0 0 |}
                                                           | 0 1 0  |  | 0 -1 0 |  | 0 0 1 |  | 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 1 0 |  | 0 0 0 |  | -1 0 0 |

o2 : LieAlgebraRepresentation

Ways to use this method:

exteriorPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\bigwedge^k V$ for a $\mathfrak{g}$-module $V$

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:2506:0.

exteriorPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\bigwedge^k V$ for a $\mathfrak{g}$-module $V$

Description

See also

Ways to use this method: