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I am a graduate student in Mathematics. I am interested in Differential Geometry and Topology. Recently I came across a new terminology namely the permanent of a square matrix.

I saw it as being defined in almost the same way as the determinant, except for the fact that the monomials are not preceded by the signatures of the corresponding permutations. The permanent of the matrix $\begin{pmatrix}a &h&g\\h &b&f\\g&f&c\end{pmatrix}$, denoted by $\operatorname{Perm} \begin{bmatrix}a &b&c\\d &e&f\\g&h&i\end{bmatrix} := aei+ahf+bdi+bfg+cdh+ceg$. However, I do not know anything about what it signifies or where it is used. Can someone help me understand the significance of the permanent of a matrix and the role it plays in Mathematics and whether there are uses of permanents in Differential Geometry, Topology and Analysis?

Thanks in Advance!

Kishalay Sarkar
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  • It's computationally distinguished by we haven't found a poly-time algorithm for computing it in certain common use cases (over what rings?), while there's the obvious algorithms for computing the determinant which works in $O(n^3)$ time. So of course many algorithms that try to compute the permanent rely on smaller (or larger) sub (sup) determinants in their intermediate algorithm steps. – Daniel Donnelly Jun 17 '25 at 05:16
  • One thing about working with the permanent is that it doesn't satisfy same things as $\det$ such as $(\det A)(\det B) = \det AB$ – Daniel Donnelly Jun 17 '25 at 05:20
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    @DanielDonnelly What about its geometric significance and uses? – Kishalay Sarkar Jun 17 '25 at 05:48
  • It has been seen in certain physics problems. A good exercise would be for you to try to create an algorithm that computes it efficiently, then you'll verify that it indeed it is tricky. Or you might just solve it for us!? – Daniel Donnelly Jun 17 '25 at 06:39
  • It's unbelievably difficult to compute, like $O(a^n)$ which means for around a useful size of about 100 you'll be waiting until next month for even an optimal C++ program to compute it, then at around 1,000,000 = n, the sun will have burned out. Now the big question is, why the hell so much difference if you alternately change sign per the determinant formula? – Daniel Donnelly Jun 17 '25 at 06:55
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    I know of no appearance of the permanent in differential geometry. – Ted Shifrin Jun 17 '25 at 17:32
  • @TedShifrin So, how exactly does Permanent come into mathematics? I mean, how is the concept of Permanent born and which fields is it relevant for? – Kishalay Sarkar Jun 18 '25 at 10:29
  • I have never encountered it before, Kishalay. I would suggest you read the wikipedia entry, and you'll know more than I do. In fact, you should have done that before even posting your question. – Ted Shifrin Jun 18 '25 at 17:58
  • @TedShifrin I did, but I could not find direct connections to Geometry. – Kishalay Sarkar Jun 21 '25 at 13:16

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