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nLab > Latest Changes: Jucys-Murphy element

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    Starting something, to record Jucys’ computation of the eigenvalues of, essentially, the Cayley distance kernel. (Thanks to David Speyer for pointing this out.)

    But I haven’t obtained copies of any of the original references yet. This page currently goes entirely by what it says on Wikipedia at Jucys-Murphy element.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2021
    • (edited May 18th 2021)

    That 1971 article seems like it might be hard to get!

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2021
    • (edited May 18th 2021)

    For what it’s worth, the Math Review of Jucys’ 1966 article looks like it could also be it (lazy cut and paste without TeXing the maths):

    The author gives new explicit expressions for the matrix units in the group algebra of the symmetric group Sn. Let Rn be the group algebra of Sn over the field of the rational functions of n variables x1,⋯,xn with real coefficients, and define the element psr of Rn by psr=(ε+(xs−xr)−1(sr)), where ε denotes the unit element and (sr) the transposition of s and r (r,s=1,⋯,n;r≠s). In order to obtain the primitive idempotent elements, a product of 12n(n−1) elements psr is formed. The primitive idempotent elements are the values of that product for certain integral values of the variables which depend on the standard tableaux. The other matrix units are obtained in a similar way.

    Here’s the review of the 1971 article, for completeness:

    The primitive idempotents of Young’s “orthogonal” representation are factorized in a way involving only mutually commuting factors of the type (rikε+Dk) (i≤k=1,2,⋯,n), ε denoting the unity of the group and Dk the sum of transpositions (kp) with p<k. The values of the real numbers rik depend on the standard Young tableaux defining the primitive idempotent (projection operator).

    And this 1974 article of Jucys cites both the earlier works, so perhaps it may be extracted from the context of the citations which of the two is needed for the theorem.

    Added I suspect you are correct, Urs, based on my perusal of (Jucys 1974). He talks about the content of a Young diagram being the eigenvalues (well, “proper values”, which is a real flashback!) and cites his 1971 paper.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    Thanks, that’s better than nothing.

    It sounds like it is indeed the 1971 article.

    But this theorem surely must have been recorded somewhere beyond this original article and the Wikipedia page? But I can’t find it anywhere.

    v1, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    Could you send me the 1974 article? I was about to download it, but my phone battery died the moment I was authenticating my library access.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2021

    OK, will do.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2021

    Murphys’ article https://doi.org/10.1016/0021-8693(92)90045-N might also be worth looking at (no paywall here). It has some background including work related to what Jucys did. And it cites (indirectly) a paper by one Thrall, which I think is this https://doi.org/10.1215/S0012-7094-41-00852-9 (the citation is via a book on representation theory of the summetric groups by JE Rutherford, the intro of which mentions a 1941 paper by Thrall, and this is the one I found).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    Thanks. So I have expanded the citations a little, also the entry text.

    Let me highlight that – while those eigenvalues are discussed widely in the literature – what I was looking for is that “factorization of the Cayley distance kernel”

    (t+J 1)(t+J 2)⋯(t+J n)=∑σ∈Sym(n)e ln(t)⋅#cycles(σ)σ∈ℂ[Sym(n)][t] \big( t + J_1 \big) \big( t + J_2 \big) \cdots \big( t + J_n \big) \;=\; \underset{ \sigma \in Sym(n) }{\sum} e^{ ln(t) \cdot \# cycles(\sigma) } \, \sigma \;\;\;\;\; \in \mathbb{C}[Sym(n)][t]

    which the Wikipedia article attributes, unspecifically, to Jucys (“Theorem (Jucys)”).

    But I guess now this is not much of a theorem: Just multiply out and observe that this generates all permutations in their minimal-number-of-transpositions-form using this kind of factorizations of its cycles.

    diff, v3, current

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2021

    Yeah, it’s disappointing when that happens…

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 18th 2021

    Perhaps there’s something to harvest from this MO discussion.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    David R.: Sounds like we are talking past each other:

    I am doubting that the second theorem that the Wikipedia page attributes to Jucys is really something that Jucys claimed as a theorem.

    It’s a little lemma (I just typed out the proof, but now Instiki gives me a mysterious error message when trying to submit).

    It’s only with Jucys’s actual theorem – the determination of those eigenvalues of the JM elements – that this lemma becomes interesting.

    But it remains unclear whether Jucys made that connection. If he made it in his 1971 article (which we haven’t seen), then he didn’t find it worth to include in the review in his 1974 article (which we have seen).

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    I have spelled out the proof of that factorization statement here.

    diff, v5, current

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2021

    Oh, I see, yes, I think I misunderstood what you meant.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 18th 2021

    More treatment of this area here maybe with some useful references.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021
    • (edited May 18th 2021)

    If you want to go deeper into this seminormal business, let’s think about what this would do for our purpose:

    Maybe one advantage of the seminormal basis {v T} T∈sYT n\big\{v_T\big\}_{T \in sYT_n} over the basis {S i,j (λ)} λ∈Part(n)1≤i,j≤dim(S (λ)) \big\{ S^{(\lambda)}_{i,j} \big\}_{ { \lambda \in Part(n) } \atop { 1 \leq i,j \leq dim(S^{(\lambda)}) } } that we used to consider is better compatibility with the inclusions Sym(n)⊂Sym(n+1)Sym(n) \subset Sym(n+1). Somehow. Not sure yet how to make use of this.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2023

    added a pdf-link (to a Russian original) for this item:

    diff, v12, current

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