symmetricPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\operatorname{Sym}^d V$ for a $\mathfrak{g}$-module $V$

Function: symmetricPower
Usage:

symmetricPower(d,V)
Inputs:
- d, an integer,
- V, an instance of the type LieAlgebraRepresentation,
Outputs:
- W, an instance of the type LieAlgebraRepresentation,

Description

Let $\rho$ be a LieAlgebraRepresentation. Then this function computes the action of $\mathfrak{g}$ on $W = \operatorname{Sym}^d V$.

In the example below, we compute $\operatorname{Sym}^2 V$ for the standard representation of $sl_2$.

i1 : V = standardRepresentation("A",1);

i2 : W = symmetricPower(2,V)

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

o2 : LieAlgebraRepresentation

Ways to use this method:

symmetricPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\operatorname{Sym}^d V$ for a $\mathfrak{g}$-module $V$

The source of this document is in LieAlgebraRepresentations/documentation.m2:2469:0.

symmetricPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\operatorname{Sym}^d V$ for a $\mathfrak{g}$-module $V$

Description

See also

Ways to use this method: