gtInvariantInVtensorVdual -- computes an invariant in $(V \otimes V^*)$ in the type A Gelfand-Tsetlin basis

Usage:

gtInvariantInVtensorVdual(lambda)
Inputs:
- lambda, a list,
Optional inputs:
- SaveAsFunction
Outputs:
- f, a ring element,

Description

Currently only defined and implemented for $G = SL_n$.

Let $\rho: SL_n \rightarrow GL(V)$ be a representation where $V$ is irreducible of highest weight $\lambda$. Then $\dim (V \otimes V^{*})^{SL_n} = 1$.

We have a conjectural combinatorial formula for this invariant polynomial in the Gelfand-Tsetlin basis of $V$. See https://faculty.fordham.edu/dswinarski/InvariantPolynomialsAndMukaiModels/InvariantPolynomialConjecture.pdf.

Here is an example for $SL_4$ and $V$ of highest weight $2\omega_1$.

i1 : gtInvariantInVtensorVdual({2,0,0})

     1       1       1        1        1        1        1        1      
o1 = -A B  - -A B  + -A B  + --A B  - --A B  + --A B  - --A B  + --A B  -
     8 0 9   8 1 8   8 2 7   12 3 6   12 4 5   24 5 4   24 6 3   24 7 2  
     ------------------------------------------------------------------------
      1        1
     --A B  + ---A B
     48 8 1   144 9 0

o1 : QQ[A ..B ]
         0   9

Ways to use gtInvariantInVtensorVdual:

gtInvariantInVtensorVdual(List)

For the programmer

The object gtInvariantInVtensorVdual is a method function with options.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2431:0.