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• 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

$s_\lambda \;=\; \frac{1}{n!} \underset {\sigma \in Sym(n)} {\sum} \chi^{(\lambda)}(\sigma) \cdot p_\sigma$

is discussed.

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

• 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

• 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.)

• 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