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nLab > Latest Changes: Schur function

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2021
    • (edited Apr 28th 2021)

    added pointer to Sagan’s textbook and encyclopedia article, and pointer to where in there the Frobenius formula

    s λ=1n!∑σ∈Sym(n)χ (λ)(σ)⋅p σ s_\lambda \;=\; \frac{1}{n!} \underset {\sigma \in Sym(n)} {\sum} \chi^{(\lambda)}(\sigma) \cdot p_\sigma

    is discussed.

    (I did not find it mentioned in either of Macdonald’s, James’s or Diaconis’ textbook)

    diff, v17, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 28th 2021

    Well done! I’d only seen it in Qiaochu Yuan’s post.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2021
    • (edited Apr 29th 2021)

    Yes, thanks, I had followed your links here, where the statement is recorded (out of the blue) as Def. 2 there.

    The only reference given in that post is to

    but I don’t see the formula in there either. In his other post he cites Sagan, though, and so possibly that’s again the source from which he got that formula.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2021
    • (edited Apr 29th 2021)

    I have now made fully explicit (here) all the ingredients that go into the Frobenius formula

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2021

    added (here) the equivalent description in terms of sums over semistandard Young tableaux.

    (Wrote this as a proposition under “Properties”, but c iting Sagan01, who gives this as the definition. Need to give a reference for the proof of the equivalence to the original Jacobi-style definition.)

    diff, v19, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021
    • (edited May 16th 2021)

    made more explicit (here) the formula in terms of ssYT for the case of finite number of variables

    diff, v23, current

1 to 6 of 6

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