By Alon Regev, Amitai Regev, and Doron Zeilberger
[With great assistance by Shalosh B. Ekhad]
"The numbers in Pascal's triangle satisfy, practically speaking, infinitely many identities, so it is not too surprising that we can find some surprising relationships by looking closely."
The aim of this note is to indicate that a similar statement seems to hold for the character tables of the symmetric groups S_{n}. Just as important, it is a case-study in using a computer algebra system to prove deep identities, way beyond the ability of mere humans.
Added Oct. 27, 2016: Christine Bessenrodt discovered (and proved) a lovely refinement
Χ^{λ}(μ)
where λ ranges over all hook shapes with n cells, and where μ is a fixed partition, μ_{0} (whose smallest part is larger than 1) followed by n-|μ_{0}| ones, for all such μ_{0}, with |μ_{0}| ≤ 14, the
Χ^{λ}(μ)
where λ ranges over all shapes with at most TWO rows, with n cells, and where μ is a fixed partition, μ_{0} (whose smallest part is larger than 1) followed by n-|μ_{0}| ones, for all such μ_{0}, with |μ_{0}| ≤ 14, the
Χ^{λ}(μ)
where λ ranges over all shapes with at most THREE rows, with n cells, and where μ is a fixed partition, μ_{0} (whose smallest part is larger than 1) followed by n-|μ_{0}| ones, for all such μ_{0}, with |μ_{0}| ≤ 9, the
Χ^{λ}(μ)
where λ ranges over all shapes with at most THREE rows, with n cells, and where μ is a fixed partition, μ_{0} (whose smallest part is larger than 1) followed by n-|μ_{0}| ones, for all such μ_{0}, with |μ_{0}| ≤ 7, the
Χ^{λ}(μ)
where λ ranges over all shapes with at most FOUR rows, with n cells, and where μ is a fixed partition, μ_{0} (whose smallest part is larger than 1) followed by n-|μ_{0}| ones, for all such μ_{0}, with |μ_{0}| ≤ 7, the
Χ^{λ}(μ)
where λ ranges over all shapes with at most FIVE rows, with n cells, and where μ is a fixed partition, μ_{0} (whose smallest part is larger than 1) followed by n-|μ_{0}| ones, for all such μ_{0}, with |μ_{0}| ≤ 5, the