Are you sure this makes a difference ?.

I have the impression it is always possible to construct any permutation as a product of permuatations that only permute 2 elements, and these would be "one-cycle" terms, as you call it.

Henri, I don't mean products of permutations. For instance, let A be a 3*3 matrix, then the quantity I am interested in is simply A12A23A31+A13A32A21.

Peter, my story is as follows. While working on my project Nonlinear Algebraic Topology and doing calculations, I encountered the following expression, where f is a 5*5 symmetric matrix:

f[1,2]*f[1,3]*f[2,4]*f[3,5]*f[4,5]-f[1,2]*f[1,4]*f[2,3]*f[3,5]*f[4,5]-f[1,2]*f[1,3]*f[2,5]*f[3,4]*f[4,5]+f[1,2]*f[1,5]*f[2,3]*f[3,4]*f[4,5]+f[1,3]*f[1,4]*f[2,3]*f[2,5]*f[4,5]-f[1,3]*f[1,5]*f[2,3]*f[2,4]*f[4,5]+f[1,2]*f[1,4]*f[2,5]*f[3,4]*f[3,5]-f[1,2]*f[1,5]*f[2,4]*f[3,4]*f[3,5]-f[1,3]*f[1,4]*f[2,4]*f[2,5]*f[3,5]+f[1,4]*f[1,5]*f[2,3]*f[2,4]*f[3,5]+f[1,3]*f[1,5]*f[2,4]*f[2,5]*f[3,4]-f[1,4]*f[1,5]*f[2,3]*f[2,5]*f[3,4].

Simply a result of some direct calculation. Then I looked at it and saw that, because f[i,j]=f[j,i], this is half the quantity I described in my question, for a 5*5 matrix.

Thank you for your interesting question, Igor Germanovich.

Sorry, I don't know the name you are asking for.

I played a little with the determinants, hoping that aii=0 would help ... unfortunately, it is not a sufficient condition for excluding ALL terms that contain more than one cycle. It is just a necessary condition.

My further hypothesis [hold for n=2,3,4,5] :

Thanks Peter, that you name the cycles correctly: full 5-cycles. I was get confused first days with "one-cycle terms".

I am reading your replay from yesterday [too much words, so I ignored it before] :o

Logically, the main diagonal consists necessarily of zeros.

With even/odd cycles, you are right, I think (I forgot the terminology). 

The symmetry that you propose is maybe a good idea, I did not checked it yet.

We have not yet found a name for such expressions!!! 

Peter, if n is odd, the determinant of the antisymmetric  (skew-symmetric) matrix is 0. This is not Igor's case.

Peter,

That is correct !. In fact you can see any permutation of the form

12345

abcde

where a ne 1, b ne 2, c ne 3, d ne 4 and e ne 5 as a cyclic permutation.

Just renumber a->2 p(a)->3, p(p(a))->4 and p(p(p(a)))->5, where p(x) permutation(x).

It also matches the number of terms. Igor has 12 terms, so because of the symmetry the determinant would produce 24, and this equals (5! - 4*4!), the number of terms in the determinant of the matrix with zeros on the main diagonal.

Dear Colleagues, 

Thank you for this discussion that I only today have had a time to look at (being busy with both work and celebrating some festive events).

Indeed, my term "one-cycle permutations" looks ambivalent, but you understood me!

I am preparing a paper with explanation where my 12-term expression (see comments to this question) has come from. Right now I can say that it appears in a `bosonic' case of a TQFT, whose `fermionic' case is described, especially, in my two latest papers on arXiv.

Thank you once again!

Dear Igor,

I am lost. I mean, after Peters last remark, it is quite clear there is no simple answer to your question. I mean, for a 3x3, it is the determinant of the matrix with the elements on the main diagonal zero'd. For a 5x5, there is a more obscure condition that Peter gave. However, if the dimension goes up, this extra condition becomes more complicated, and I don't see a generalized answer to this question. So, if you have it (and have just been teasing us), could you please explain us ?.

Best Regards,

Henri

Henri, thank you for your question. Please just let me have some time to write a text. Right now, I can say that it is about a bosonic parallel to my paper https://arxiv.org/abs/1310.4075 (note that something in this direction already exists: https://arxiv.org/abs/1405.5763 , but this needs, I think, more investigation in its inner structure). Also, what I have encountered as yet has to do only with 5*5 matrices. The number 5 corresponds to five 3-faces of a pentachoron.

And not that I meant to tease innocent people (although why not?), I rather thought it plausible that some of you may have encountered such structures somewhere. Please believe that my question is actually teasing for me as well :D

Dear Colleagues,

I have now written the text explaining where I took my "cyclic" expression, see Eq. (11) at http://www.researchgate.net/publication/316617653

Briefly, it appears when I reduce a Gaussian exponential to what may be called a canonical form. And the quadratic form in the exponent depends on five variables living on 3-faces of a pentachoron. And to each of these variables corresponds a `gauge group' Sp(2,C).

Article Bosonic pentachoron weights and multiplicative 2-cocycles