<p>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</p>

<p>I have been going through Jin-Yi Cai's <a href="http://pages.cs.wisc.edu/~jyc/papers/matchgate-arts-to-sc.pdf" rel="nofollow">expository paper</a> on Holographic algorithms.</p>

<p>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.</p>

<p>My question arises from the following statement.</p>

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

<p>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.</p>

<p>I would be glad if anyone could help me with this.</p>

<p>Thanks
-Akash</p>

CC BY-SA 2.5
2011-01-13T17:25:34.353Z
http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors, and also this is more appropriate for math.SE
Suresh Venkat
CC BY-SA 2.5
2011-01-13T18:23:26.200Z
@suresh - I went to the same url. Unfortunately, they do not talk about "type" of contravariant tensors
Akash Kumar
CC BY-SA 2.5
2011-01-13T19:58:38.377Z
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
Suresh Venkat
<terminology><tensor-rank>
What are contravariant tensors of type m choose 0
Akash Kumar
# What are contravariant tensors of type m choose 0

(Asked by: username_0 on 2011-01-13)

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](http://pages.cs.wisc.edu/~jyc/papers/matchgate-arts-to-sc.pdf) 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

**Tags:** terminology, tensor-rank
> **Comments:**
> **username_1**: http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors, and also this is more appropriate for math.SE  
> **username_0**: @suresh - I went to the same url. Unfortunately, they do not talk about "type" of contravariant tensors  
> **username_1**: 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  

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## 0 answers

