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 ℙ(S^dCⁿ⁺¹) of symmetric forms of degree d in n + 1 variables which are equivariant for the action of SL_n+1(C) ,and admit an equivariant free resolution of length 2.

Take two integers d ≥1 and m ≥2 and a vector spave V = Cⁿ⁺¹. For n=2, we have

S^dV ⊗S^(m-1)dV = S^mdV ⊕S^md-2V ⊕S^md-4V ⊕…,

while for n > 1,

S^dV ⊗S^(m-1)dV = S^mdV ⊕V_{(md-2)λ1+λ2} ⊕V_{(md-4)λ1+2λ2} ⊕…,

where λ₁ and λ₂ are the two greatest fundamental weights of the Lie group SL_n+1(C) and V_iλ1+jλ2 is the irreducible representation of highest weight iλ₁+jλ₂.

The projection of the tensor product onto the second summand induces a SL₂(C)-equivariant morphism

Φ: S^md-2V ⊗Oℙ(S^dV) →S^(m-1)dV ⊗Oℙ(S^dV)(1)

or a SL_n+1(C)-equivariant morphism

Φ: V_{(md-2)λ1 + λ2} ⊗Oℙ(S^dV) →S^(m-1)dV ⊗Oℙ(S^dV)(1)

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

0 →W_2,d,m →S^md-2V ⊗Oℙ(S^dV) →S^(m-1)dV ⊗Oℙ(S^dV)(1) →Oℙ(S^dV)(m) →0,

0 →W_n,d,m →V_{(md-2)λ1 + λ2} ⊗Oℙ(S^dV) →S^(m-1)dV ⊗Oℙ(S^dV)(1) →Oℙ(S^dV)(m) →0.

The package allows to compute

(1) the decomposition into irreducible SL_n+1(C)-representations of the tensor product of two symmetric powers S^aCⁿ⁺¹ and S^bCⁿ⁺¹;

(2) the matrix representing the morphism Φ;

(3) the vector bundle W_n,d,m.