The negative $σ$-moment generating function
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918049543618560 |
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| author | Howe, Sean |
| author_facet | Howe, Sean |
| contents | For $X$ a pre-$λ$ random variable, we show the $σ$-moment generating function of $-X$ can be obtained from the $σ$-moment generating function of $X$ by applying the composition of the standard and degree flip involutions on symmetric power series. This isometric involution is natural as it preserves the pre-$λ$ ring structure on symmetric power series with pre-$λ$ coefficients, thus this formula provides a simple description of the $σ$-moment generating function of $-X$ whenever the $σ$-moment generating function of $X$ has a simple description using the pre-$λ$ structure. As an application we compute, in a natural range, the dimensions of orthogonal and symplectic group invariants in tensor products of exterior powers of their standard representations on $\mathbb{C}^n$. We also compute a generating function for stable traces of Frobenius related to the moment conjecture for prime-order function field Dirichlet characters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The negative $σ$-moment generating function Howe, Sean Representation Theory Number Theory For $X$ a pre-$λ$ random variable, we show the $σ$-moment generating function of $-X$ can be obtained from the $σ$-moment generating function of $X$ by applying the composition of the standard and degree flip involutions on symmetric power series. This isometric involution is natural as it preserves the pre-$λ$ ring structure on symmetric power series with pre-$λ$ coefficients, thus this formula provides a simple description of the $σ$-moment generating function of $-X$ whenever the $σ$-moment generating function of $X$ has a simple description using the pre-$λ$ structure. As an application we compute, in a natural range, the dimensions of orthogonal and symplectic group invariants in tensor products of exterior powers of their standard representations on $\mathbb{C}^n$. We also compute a generating function for stable traces of Frobenius related to the moment conjecture for prime-order function field Dirichlet characters. |
| title | The negative $σ$-moment generating function |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2505.01205 |