Lie group GL(4) representation decomposition Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$.
Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation.
My question: How does $$V\otimes Ext\; V$$ decompose into irreducibles?
I am especially interested in the multiplicity of $V$ and its contragredient. Root system: $A_3$ I fail to visualize like $A_2$.
 A: This is an example in Fulton and Harris's 
Representation Theory: A First Course (Springer GTM 129 (1991)),
except that they work only with $SL(V)$, not $GL(V)$.
The representation you call $Ext\;V$ is denoted $\wedge^2 V$.
Its tensor product with $V$ has a natural map to $\wedge^3 V$,
which is isomorphic with the contragredient $V^*$ as a representation 
of $SL(V)$, but not quite for $GL(V)$ (where $\wedge^3 V$ is the 
tensor product of $V^*$ with the 1-dimensional determinant
representation).  For the kernel of this map, Fulton and Harris
say on page 220 that it contains "the irreducible representation
$\Gamma_{1,1,0}$ with highest weight $2L_1+L_2$", and then Exercise 15.10
asserts that the kernel is in fact this irreducible representation.
So in summary Fulton and Harris say that $V \otimes \wedge^2 V$ 
is the direct sum of irreducible representations of dimensions $4$ and $20$,
the former of which is $\wedge^3 V = V^* \otimes \det$.
P.S. The Fulton-Harris text also shows pictures of 
the root system etc. for $A_3$.
