VInWedgekW -- computes a basis of a submodule of $\bigwedge^k W$ isomorphic to $V$ with a given highest weight vector

Usage:

VInWedgekW(V,k,W,hwv)
Inputs:
- V, an instance of the type LieAlgebraRepresentation,
- k, an integer,
- W, an instance of the type LieAlgebraRepresentation,
- hwv, a matrix,
Optional inputs:
- SaveAsFunction
Outputs:
- L, a list,

Description

Suppose that an irreducible module $V$ appears in the decomposition of $\bigwedge^k W$ with multiplicity at least one. Then we can find a highest weight vector using weightMuHighestWeightVectorsInW, and then compute a basis of a submodule in $\bigwedge^k W$ isomorphic to $V$. The basis elements are expressed as linear polynomials in the Plücker coordinates.

In the example below, let $U$ be the standard representation for $sl_4$, and let $W = \bigwedge^2 U$. Then $\bigwedge^3 W$ contains an irreducible submodule with highest weight $(2,0,0)$. We compute a basis for this submodule explicitly.

i1 : sl4=simpleLieAlgebra("A",3);

i2 : U= standardRepresentation(sl4);

i3 : W2U = exteriorPower(2,U);

i4 : W3W2U = exteriorPower(3,W2U);

i5 : hwv = weightMuHighestWeightVectorsInW({2,0,0},W3W2U)

o5 = | 0 |
     | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |

              20       1
o5 : Matrix QQ   <-- QQ

i6 : V=irreducibleLieAlgebraModule({2,0,0},sl4);

i7 : LAB = lieAlgebraBasis(sl4);

i8 : V = lieAlgebraRepresentation(V,LAB,GTrepresentationMatrices(V));

i9 : L = VInWedgekW(V,3,W2U,hwv)
Length 1 complete. 3 new words found
Length 2 complete. 6 new words found

o9 = {p         , p          + p         , 2p         , 3p          + 3p   
       {0, 1, 3}   {0, 2, 3}    {0, 1, 4}    {0, 2, 4}    {1, 2, 3}     {0,
     ------------------------------------------------------------------------
          , 3p          + 3p         , 12p         , - 12p          + 12p   
     1, 5}    {1, 2, 4}     {0, 2, 5}     {1, 2, 5}       {1, 3, 4}      {0,
     ------------------------------------------------------------------------
          , - 12p          + 12p         , - 24p          + 24p         ,
     3, 5}       {2, 3, 4}      {0, 4, 5}       {2, 3, 5}      {1, 4, 5} 
     ------------------------------------------------------------------------
     144p         }
         {3, 4, 5}

o9 : List

Ways to use VInWedgekW:

VInWedgekW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,Matrix)

For the programmer

The object VInWedgekW is a method function with options.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2859:0.