# What are contravariant tensors of type m choose 0

I am not sure if the following question falls within the scope of this site; if it does not, I will request the moderators to take appropriate action

I have been going through Jin-Yi Cai's expository paper on Holographic algorithms.

I had a question. Immediately after introducing Grassman-Plucker Identities for Pfaffians and the definitions for matchgates, recognizers and generators, Cai introduces signature tensor for matchgates.

My question arises from the following statement.

He states that a generator $\Gamma$ with $m$ output nodes is assigned a contravariant tensor $\mathbf{G} \in V_0^m$ of type $\binom {m} {0}$.

I looked up contravariant tensors on wikipedia but did not find any reference to the type of contravariant tensors which Cai talks about. Maybe I am not searching properly, but I was unable to find a proper reference which defines what I am looking for.

I would be glad if anyone could help me with this.

Thanks -Akash

• en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors, and also this is more appropriate for math.SE Jan 13 '11 at 17:25
• @suresh - I went to the same url. Unfortunately, they do not talk about "type" of contravariant tensors Jan 13 '11 at 18:23
• the type is the number of terms in the tensor product. Informally, a contravariant tensor of type m corresponds to the wedge product of m vectors Jan 13 '11 at 19:58