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

## Description

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(SdCn+1) of symmetric forms of degree d in n + 1 variables which are equivariant for the action of SLn+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

SdV ⊗S(m-1)dV = SmdV ⊕Smd-2V ⊕Smd-4V ⊕…,

while for n > 1,

SdV ⊗S(m-1)dV = SmdV ⊕V(md-2)λ12 ⊕V(md-4)λ1+2λ2 ⊕…,

where λ1 and λ2 are the two greatest fundamental weights of the Lie group SLn+1(C) and V1+jλ2 is the irreducible representation of highest weight 1+jλ2.

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

Φ: Smd-2V ⊗Oℙ(SdV) →S(m-1)dV ⊗Oℙ(SdV)(1)

or a SLn+1(C)-equivariant morphism

Φ: V(md-2)λ1 + λ2 ⊗Oℙ(SdV) →S(m-1)dV ⊗Oℙ(SdV)(1)

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

0 →W2,d,m →Smd-2V ⊗Oℙ(SdV) →S(m-1)dV ⊗Oℙ(SdV)(1) →Oℙ(SdV)(m) →0,

0 →Wn,d,m →V(md-2)λ1 + λ2 ⊗Oℙ(SdV) →S(m-1)dV ⊗Oℙ(SdV)(1) →Oℙ(SdV)(m) →0.

The package allows to compute

(1) the decomposition into irreducible SLn+1(C)-representations of the tensor product of two symmetric powers SaCn+1 and SbCn+1;

(2) the matrix representing the morphism Φ;

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

## Version

This documentation describes version 1.0 of SLnEquivariantMatrices.

## Source code

The source code from which this documentation is derived is in the file SLnEquivariantMatrices.m2.