SLnEquivariantMatrices -- Ancillary file to the paper "A construction of equivariant bundles on the space of symmetric forms"

In the paper "A construction of equivariant bundles on the space of symmetric forms" (https://arxiv.org), the authors construct stable vector bundles on the space $\PP(S^d\CC^{n+1})$ of symmetric forms of degree $d$ in $n + 1$ variables which are equivariant for the action of $SL_{n+1}(\CC)$ ,and admit an equivariant free resolution of length 2.

Take two integers $d \ge 1$ and $m \ge 2$ and a vector spave $V = \CC^{n+1}$. For $n=2$, we have

$S^dV \otimes S^{(m-1)d}V = S^{md}V \oplus S^{md-2}V \oplus S^{md-4}V \oplus \dots$,

while for $n > 1$,

$S^dV \otimes S^{(m-1)d}V = S^{md}V \oplus V_{(md-2)\lambda_1+\lambda_2} \oplus V_{(md-4)\lambda_1+2\lambda_2} \oplus \dots$,

where $\lambda_1$ and $\lambda_2$ are the two greatest fundamental weights of the Lie group $SL_{n+1}(\bf C)$ and $V_{i\lambda_1+j\lambda_2}$ is the irreducible representation of highest weight $i\lambda_1+j\lambda_2$.

The projection of the tensor product onto the second summand induces a $SL_{2}(\CC)$-equivariant morphism

$\Phi: S^{md-2}V \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V \otimes O_{\PP(S^dV)}(1)$

or a $SL_{n+1}(\CC)$-equivariant morphism

$\Phi: V_{(md-2)\lambda_1 + \lambda_2} \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V \otimes O_{\PP(S^dV)}(1)$

with constant co-rank 1, and thus gives an exact sequence of vector bundles on $\PP(S^dV)$:

$0 \to W_{2,d,m} \to S^{md-2}V \otimes \mathcal{O}_{\PP(S^dV)} \to S^{(m-1)d}V \otimes \mathcal{O}_{\PP(S^dV)}(1) \to \mathcal{O}_{\PP(S^dV)}(m) \to 0$,

$0 \to W_{n,d,m} \to V_{(md-2)\lambda_1 + \lambda_2} \otimes \mathcal{O}_{\PP(S^dV)} \to S^{(m-1)d}V \otimes \mathcal{O}_{\PP(S^dV)}(1) \to \mathcal{O}_{\PP(S^dV)}(m) \to 0$.

The package allows to compute

(1) the decomposition into irreducible $SL_{n+1}(\bf C)$-representations of the tensor product of two symmetric powers $S^a\CC^{n+1}$ and $S^b\CC^{n+1}$;

(2) the matrix representing the morphism $\Phi$;

(3) the vector bundle $W_{n,d,m}$.