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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2412.18368 |
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| _version_ | 1866917887265996800 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | Given an identity relating families of Schur and power sum symmetric functions, this may be thought of as encoding representation-theoretic properties according to how the $p$-to-$s$ transition matrices provide the irreducible character tables for symmetric groups. The case of the Murnaghan-Nakayama rule for cycles provides that $p_{n} = \sum_{i = 0}^{n-1} (-1)^i s_{(n-i, 1^{i})}$, and, since the power sum generator $p_{n}$ reduces to $ζ(2n)$ for the Riemann zeta function $ζ$ and for specialized values of the indeterminates involved in the inverse limit construction of the algebra of symmetric functions, this motivates both combinatorial and number-theoretic applications related to the given case of the Murnaghan-Nakayama rule. In this direction, since every Schur-hook admits an expansion in terms of twofold products of elementary and complete homogeneous generators, we exploit this property for the same specialization that allows us to express $p_{n}$ with the Bernoulli number $B_{2n}$, using remarkable results due to Hoffman on multiple harmonic series. This motivates our bijective approach, through the use of sign-reversing involutions, toward the determination of identities that relate Schur-hooks and power sum symmetric functions and that we apply to obtain a new recurrence for Bernoulli numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schur-hooks and Bernoulli number recurrences Campbell, John M. Combinatorics 05E05 Given an identity relating families of Schur and power sum symmetric functions, this may be thought of as encoding representation-theoretic properties according to how the $p$-to-$s$ transition matrices provide the irreducible character tables for symmetric groups. The case of the Murnaghan-Nakayama rule for cycles provides that $p_{n} = \sum_{i = 0}^{n-1} (-1)^i s_{(n-i, 1^{i})}$, and, since the power sum generator $p_{n}$ reduces to $ζ(2n)$ for the Riemann zeta function $ζ$ and for specialized values of the indeterminates involved in the inverse limit construction of the algebra of symmetric functions, this motivates both combinatorial and number-theoretic applications related to the given case of the Murnaghan-Nakayama rule. In this direction, since every Schur-hook admits an expansion in terms of twofold products of elementary and complete homogeneous generators, we exploit this property for the same specialization that allows us to express $p_{n}$ with the Bernoulli number $B_{2n}$, using remarkable results due to Hoffman on multiple harmonic series. This motivates our bijective approach, through the use of sign-reversing involutions, toward the determination of identities that relate Schur-hooks and power sum symmetric functions and that we apply to obtain a new recurrence for Bernoulli numbers. |
| title | Schur-hooks and Bernoulli number recurrences |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2412.18368 |