# Decomposition of product of exterior products

Suppose $V$ is a finite dimensional vector space of dimension n.

What is the kernel of the map

$$\bigwedge^p V \otimes \bigwedge^q V ----> \bigwedge^{p+q} V$$ ?

(here $p+q< n$)

Thanks.. Jyoti

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This question would be a better fit at math.stackexchange.com - see mathoverflow.net/faq#whatquestions – David Roberts Aug 16 '11 at 6:56
I can't give you an answer, but if you look at Fulton and Harris, Representation Theory, chapter 15, you will find the general method to decompose representations of $GL(V)$ into irreducibles. – Ben McKay Aug 16 '11 at 8:25
Not sure why this was closed. Anyway, there is an elegant answer. Assuming you understand the classification of irreps of GL(V) by partitions, Pieri's rule tells you that $\bigwedge^p V \otimes \bigwedge^q V \cong \bigoplus S_{2^a 1^b} V$, where $2a+b=p+q$ and $a \leq \min(p,q)$. The map to $\bigwedge^{p+q} V$ is projection onto the $(a,b)=(0,p+q)$ summand, so the kernel is the direct sum of the other components. – David Speyer Aug 16 '11 at 12:02
I voted to reopen. I agree with David -- I think people jumped the gun on voting to close it. – Andy Putman Aug 16 '11 at 14:56
Hmm, on a 2nd look I think I misread/misunderstood the question, though I still think it could do with added context for why the question is being asked. Anyway, voting to re-open (volte-face) – Yemon Choi Aug 16 '11 at 17:56