Lower Bound for The Number of Zeros in The Character Table of The Symmetric Group
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911554636611584 |
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| author | Barman, Jayanta Mahatab, Kamalakshya |
| author_facet | Barman, Jayanta Mahatab, Kamalakshya |
| contents | For any two partitions $λ$ and $μ$ of a positive integer $N$, let $χ_λ(μ)$ be the value of the irreducible character of the symmetric group $S_{N}$ associated with $λ$, evaluated at the conjugacy class of elements whose cycle type is determined by $μ$. Let $Z(N)$ be the number of zeros in the character table of $S_N$, and $Z_{t}(N)$ be defined as
$$
Z_{t}(N):= \#\{(λ,μ): χ_λ(μ) = 0 \; \text{with $λ$ a $t$-core}\}.
$$
We prove
$$
Z(N) \ge \frac{2\, p(N)^{2}}{\log N} \left(1+O\left(\frac{1}{\sqrt{\log N}}\right)\right),
$$
where $p(N)$ denotes the number of partitions of $N$. We also give explicit lower bounds for $Z_t(N)$ in various ranges of $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17037 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower Bound for The Number of Zeros in The Character Table of The Symmetric Group Barman, Jayanta Mahatab, Kamalakshya Number Theory Combinatorics Representation Theory 20C30, 11P82, 05A17 For any two partitions $λ$ and $μ$ of a positive integer $N$, let $χ_λ(μ)$ be the value of the irreducible character of the symmetric group $S_{N}$ associated with $λ$, evaluated at the conjugacy class of elements whose cycle type is determined by $μ$. Let $Z(N)$ be the number of zeros in the character table of $S_N$, and $Z_{t}(N)$ be defined as $$ Z_{t}(N):= \#\{(λ,μ): χ_λ(μ) = 0 \; \text{with $λ$ a $t$-core}\}. $$ We prove $$ Z(N) \ge \frac{2\, p(N)^{2}}{\log N} \left(1+O\left(\frac{1}{\sqrt{\log N}}\right)\right), $$ where $p(N)$ denotes the number of partitions of $N$. We also give explicit lower bounds for $Z_t(N)$ in various ranges of $t$. |
| title | Lower Bound for The Number of Zeros in The Character Table of The Symmetric Group |
| topic | Number Theory Combinatorics Representation Theory 20C30, 11P82, 05A17 |
| url | https://arxiv.org/abs/2504.17037 |