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.