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Foulkes Conjecture

Foulkes' conjecture asserts that for integers 1<=a<=b and a finite-dimensional complex vector space V, there is a GL(V)-equivariant injection

Sym^a(Sym^bV)->Sym^b(Sym^aV).

Equivalently, the difference of the characters, written as the plethysm h_b[h_a]-h_a[h_b], is Schur positive. Here h_a is the complete homogeneous symmetric polynomial. The assertion requires an injection but does not specify a canonical map.

See also

Complete Homogeneous Symmetric Polynomial, Foulkes-Howe Map, Plethysm, Schur Polynomial, Symmetric Power

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References

Ikenmeyer, C. "On McKay's Propagation Theorem for the Foulkes Conjecture." 16 Sep 2015. https://arxiv.org/abs/1509.04957.Sagan, B. E. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd ed. New York: Springer, 2001. https://doi.org/10.1007/978-1-4757-6804-6.

Cite this as:

Weisstein, Eric W. "Foulkes Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FoulkesConjecture.html

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