VInSymdW -- computes a basis of a submodule of $\operatorname{Sym}^d W$ isomorphic to $V$ with a given highest weight vector

Usage:

VInSymdW(V,d,W,hwv)
Inputs:
- V, an instance of the type LieAlgebraRepresentation,
- d, 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 $\operatorname{Sym}^d W$ with multiplicity at least one. Then we can find a highest weight vector using weightMuHighestWeightVectorsInSymdW, and then compute a basis of a submodule in $\operatorname{Sym}^d W$ isomorphic to $V$. The basis elements are expressed as polynomials in the basis of $W$ used to define the matrix generators of the representation on $W$.

We compute the degree four invariant for plane cubics by finding a trivial submodule in $\operatorname{Sym}^4 \operatorname{Sym}^3 \mathbb{C}^3$.

i1 : sl3=simpleLieAlgebra("A",2)

o1 = sl3

o1 : simple LieAlgebra

i2 : V=standardRepresentation(sl3);

i3 : S3V = symmetricPower(3,V);

i4 : hwv = weightMuHighestWeightVectorsInSymdW({0,0},4,S3V);
Constructing the Casimir operator...
Other EVs: {120, 76, 60, 48, 36, 30, 24, 16}
Beginning projections...
    j=0:
        EV 120 complete
        EV 76 complete
        EV 60 complete
        EV 48 complete
        EV 36 complete
        EV 30 complete
        EV 24 complete
        EV 16 complete
    #hwvs=0

i5 : V0=trivialRepresentation(sl3);

i6 : L = VInSymdW(V0,4,S3V,hwv_0)
Length 1 complete. 0 new words found
0

                      2                                         2    2      
o6 = {B B B B  - B B B  - B B B B  + B B B B  + B B B B  - B B B  - B B B  +
       0 3 7 9    0 3 8    0 4 6 9    0 4 7 8    0 5 6 8    0 5 7    1 7 9  
     ------------------------------------------------------------------------
      2 2                                                   2                
     B B  + B B B B  - B B B B  + B B B B  - B B B B  - 2B B B  + 3B B B B  -
      1 8    1 2 6 9    1 2 7 8    1 3 4 9    1 3 5 8     1 4 8     1 4 5 7  
     ------------------------------------------------------------------------
        2      2        2 2      2                                2    
     B B B  - B B B  + B B  - B B B  + 3B B B B  - B B B B  - 2B B B  +
      1 5 6    2 6 8    2 7    2 3 9     2 3 4 8    2 3 5 7     2 4 7  
     ------------------------------------------------------------------------
                 2 2       2      4
     B B B B  + B B  - 2B B B  + B }
      2 4 5 6    3 5     3 4 5    4

o6 : List

This polynomial appears as early as 1856 in work of Cayley, who attributes it to Salmon. See Cayley, "A third memoir upon quantics", tables 62 and 63.

Ways to use VInSymdW:

VInSymdW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,Matrix)
VInSymdW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,RingElement) (missing documentation)

For the programmer

The object VInSymdW is a method function with options.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2812:0.