The Frobenius transform of a symmetric function
Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 931-958.

We define an abelian group homomorphism $ℱ$, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $ℱ$ in the Schur basis are the restriction coefficients ${r}_{\lambda }^{\mu }=dim{Hom}_{{𝔖}_{n}}\left({V}_{\mu },{𝕊}^{\lambda }{ℂ}^{n}\right)$, which are known to be nonnegative integers but have no known combinatorial interpretation. The Frobenius transform satisfies the identity $ℱ\left\{fg\right\}=ℱ\left\{f\right\}*ℱ\left\{g\right\}$, where $*$ is the Kronecker product.

We prove for all symmetric functions $f$ that $ℱ\left\{f\right\}={ℱ}_{\mathrm{Sur}}\left\{f\right\}·\left(1+{h}_{1}+{h}_{2}+\cdots \right)$, where ${ℱ}_{\mathrm{Sur}}\left\{f\right\}$ is a symmetric function with the same degree and leading term as $f$. Then, we compute the matrix entries of ${ℱ}_{\mathrm{Sur}}$ in the complete homogeneous, elementary, and power sum bases and of ${ℱ}_{\mathrm{Sur}}^{-1}$ in the complete homogeneous and elementary bases, giving combinatorial interpretations of the coefficients where possible. In particular, the matrix entries of ${ℱ}_{\mathrm{Sur}}^{-1}$ in the elementary basis count words with a constraint on their Lyndon factorization.

As an example application of our main results, we prove that ${r}_{\lambda }^{\mu }=0$ if $|\lambda \cap \stackrel{^}{\mu }|<2|\stackrel{^}{\mu }|-|\lambda |$, where $\stackrel{^}{\mu }$ is the partition formed by removing the first part of $\mu$. We also prove that ${r}_{\lambda }^{\mu }=0$ if the Young diagram of $\mu$ contains a square of side length greater than ${2}^{{\lambda }_{1}-1}$, and this inequality is tight.

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Accepted:
Published online:
DOI: 10.5802/alco.365
Classification: 05E05
Keywords: symmetric functions, representation theory of categories, plethysm, Kronecker product

Lee, Mitchell 1

1 Harvard University Dept. of mathematics 1 Oxford st. Cambridge MA 02138 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Lee, Mitchell. The Frobenius transform of a symmetric function. Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 931-958. doi : 10.5802/alco.365. https://alco.centre-mersenne.org/articles/10.5802/alco.365/

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