It’s all about permutations

Posted on 2025/12/28 by arkajad

Permutations appear whenever “things are the same, but in a different order.” That simple idea hides a powerful engine that underlies physics, geometry, information, and even the notion of identity itself.

In quantum mechanics, identical particles do not carry little name tags. Two electrons swapped in space give a state that is “the same up to a phase.” The swap is a permutation, and the phase is a one-dimensional representation of the permutation group on two letters.

For bosons, permutations act trivially: exchange two bosons and the wavefunction is unchanged.

For fermions, odd permutations contribute a minus sign, and the Pauli exclusion principle is encoded in this antisymmetry.

More exotic anyons in two dimensions correspond to richer representations of braid groups, which generalize permutations by letting trajectories wind around each other.

Thus whole classes of matter are distinguished by how their quantum states transform under permutations of identical particles. The statistics of the universe are, at base, choices about how permutation symmetry is represented.

In topology, a covering space has fibers over each point; going around a loop downstairs lifts to a permutation of the fiber upstairs. The resulting homomorphism from the fundamental group to a permutation group is the monodromy.

in differential geometry, the orientation of a frame is encoded in whether the change of basis is an even or odd permutation of a fixed oriented basis, wrapped in a continuous transformation.

Permutations are not just a toy in elementary group theory; they are the algebraic shadow of a deeper idea: that the universe is less about what things and relations are, and more about how indistinguishable pieces can be rearranged to create complexity.

In the future we will use a bunch of classical formulas often used when dealing with antisymmetric tensors (multivectors and differential forms). They can be easily found online, but here I am simply sharing just two pages from “Mathematical Handbook For Scientists And Engineers“, Granino A. Korn and Theresa M. Korn, Dover 2000 (or: Г. Корн, Т. Корн, “Справочник По Математике Для Научных Работников И Инженеров”, «Наука» 1973)Some formulas are better in the Russian translation/edition:

Exercise 1. I asked Perplexity AI to write for me all properties of deltas and epsilons. One of the formulas Perplexity wrote was this one:
If a metric g_{ij} exists,

(1)   \begin{equation*} \varepsilon^{i_1 \dots i_n} = g^{i_1 j_1} \cdots g^{i_n j_n} \varepsilon_{j_1 \dots j_n}. \end{equation*}

Is it correct? If not, what should be written instead?

Note. Formula 16.5-4, as it is written in the English Dover edition is WRONG! Why?

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10 Responses to It’s all about permutations

  1. Anna says:
    2025/12/30 at 20:52

    Ark, how do you know my latest hit idea? It’s specifically about permutations. Sergei Vekhshenov and I were talking about this just a few days ago in connection with infinities. What do you get if you add 1 to infinity? Infinity again. But something has changed—what exactly? It’s the order of the elements that make up the infinity: the newly added element has replaced the previous one. And that’s a permutation! This situation hints at the possibility of another infinity—one that’s not quantitative. And it’s all about permutations, right… Permutations come into play when quantities are powerless. They exist at some higher level above numbers, and this two-level structure seems to be sufficient for everything.

    Korn, T. Korn, “Mathematical Handbook For Scientists And Engineers,” Nauka, 1973) was our reference book—mine and my husband’s. Now we have two copies of this book gathering dust somewhere in our country house… But I will gladly return to it and even try to think about exercise 1.

  2. Anna says:
    2026/01/02 at 19:42

    Indeed, I didn’t like these two formulas the moment I saw them:
    2) = 0 1
    3) = 0 -1
    I think the zeros should be removed.
    Thus, according to these formulas, epsilon with subscripts is equal to epsilon with superscripts, and the metric is simply unity.
    But I don’t know what this would look like if the metric were not Euclidean.

    • arkajad says:
      2026/01/03 at 08:37

      So, what should be on the left of (1)? Can you find the answer adapting Korn’s formulas?

      • Anna says:
        2026/01/03 at 11:00

        Frankly speaking, I don’t see an error in (1). According to (1)–(3) in 16.5-3, epsilons with subscripts are always equal to the corresponding epsilons with superscripts, and they are simultaneously equal to 0, -1, or 1 depending on the combinations of indices. Meanwhile, metrics g is a universal method to raise/lower indices and it can be used regardless of the value of epsilon. What is wrong here?

        • arkajad says:
          2026/01/03 at 11:06

          I think that if use 16.5-10, we should get the an extra factor

          \det(g^{-1} on the left hand side. For an Euclidean g this would be 1, but for a general $g# it should be there.

          • Anna says:
            2026/01/03 at 13:43

            Yes, I looked at formula 16.5-10, it seemed to be promising for the case. Now, when I know the answer, I will try to convince myself that it is really so. At least I feel those scaring formulas become a little more familiar. Thank you!

  3. Anna says:
    2026/01/05 at 09:46

    Ark, I am sorry for returning once again to the subject, but I am in doubt:

    According to (16.5-7) epsilon with upper indices is related to epsilon with lower indices precisly like deltas with subscripts and superscripts being interchanged, right?

    In compliance with the definition of delta-object in 16.5-2,
    delta with i_r superscripts and k_r subscripts is equal to +1 or -1 if
    “the ordered set of subscripts is obtained by an even or odd permutation of the set of superscripts”, and is zero for all other combinations.
    This means that delta with i_r superscripts and k_r subscripts and delta with k_r superscripts and i_r subscripts take the same values simultaneously, don’t they?

    So, epsilons with upper and lower indices appear to be the same anyway!

    I feel that metrics should emerge somewhere, this is surely reasonable, but could you show how it happens rigorously?

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