Modern Algebra II Spring 2015. Homework No.2

1. Let V,W be two vector spaces. Prove that the natural action of GL(V )×GL(W ) on

V ⊗W has finitely many orbits.

Hint: Prove that the tensors corresponding to matrices of fixed rank form one orbit.

2. Let V,W be two vector spaces of dimensions m,n respectively. Let t ∈ V ⊗W . What

is the minimal number r such that t can be expresed as a sum of r decomposable

tensors ?

3. Let V be a vector space of dimension 2. The group GL(V ) acts of the symmetric

power S4V . Prove that this action has infinitely many orbits.

4. Let us define the linear map

φ : V ∗ ⊗i∧V →

i−1∧V

by the formula

l ⊗ v1 ∧ . . . ∧ vi 7→i∑

s=1

(−1)s−1l(vs)v1 ∧ . . . ∧ v̂s ∧ . . . ∧ vs

Prove that the map φ is well-defined and does not depend on the choice of basis.

5. Try to generalize problem 4 to define the linear map

φ :

j∧V ∗ ⊗

i∧V →

i−j∧V

for all i ≥ j.6. Let v ∈ V . Describe the set of all the tensors t ∈

∧iV such that t ∧ v = 0 in

∧i+1V .