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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. – EffingLoveMath 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 $(\det A)(\det B) = \det AB$ – EffingLoveMath 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!? – EffingLoveMath 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? – EffingLoveMath 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

1 Answers1

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The analogy to the determinant is, IMO, very misleading. The permanent is really not similar to the determinant at all, for a fundamental reason: it is not invariant under change of coordinates! The determinant is really a property of a linear transformation $T : V \to V$ from a finite-dimensional vector space to itself, and can be calculated by writing down the matrix of $T$ with respect to any basis; nothing like this is true of the permanent.

In fact we know exactly what the polynomial invariants of a square matrix $M$ are, in the sense that we know exactly which polynomial functions of the entries of a square matrix are invariant under change of coordinates: they are exactly polynomials in the coefficients of the characteristic polynomial $\det(\lambda I - M)$, or equivalently, exactly the symmetric polynomials in the eigenvalues of $M$. So the determinant is really fundamental to linear algebra in a way that the permanent is not.

This means the permanent does not really belong to linear algebra, in some sense; as Wikipedia says it more belongs to combinatorics. The Wiki article has some more information about applications of the permanent.

Qiaochu Yuan
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