Equality of skew Schur functions in noncommuting variables
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909473900068864 |
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| author | Jin, Emma Yu van Willigenburg, Stephanie |
| author_facet | Jin, Emma Yu van Willigenburg, Stephanie |
| contents | The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting variables, $s_{(δ,D)}$, where $D$ is a connected skew diagram with $n$ boxes and $δ$ is a permutation in the symmetric group $S_n$.
In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: $s_{(δ,D)} = s_{(τ,T)}$ such that $D\ne T$ if and only if $D$ is a nonsymmetric ribbon, $T$ is the antipodal rotation of $D$ and $\overline{τ^{-1}δ}$ is an explicit bijection between two set partitions determined by $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19744 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equality of skew Schur functions in noncommuting variables Jin, Emma Yu van Willigenburg, Stephanie Combinatorics Primary 05E05, Secondary 05A05, 05A18, 16T30 The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting variables, $s_{(δ,D)}$, where $D$ is a connected skew diagram with $n$ boxes and $δ$ is a permutation in the symmetric group $S_n$. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: $s_{(δ,D)} = s_{(τ,T)}$ such that $D\ne T$ if and only if $D$ is a nonsymmetric ribbon, $T$ is the antipodal rotation of $D$ and $\overline{τ^{-1}δ}$ is an explicit bijection between two set partitions determined by $D$. |
| title | Equality of skew Schur functions in noncommuting variables |
| topic | Combinatorics Primary 05E05, Secondary 05A05, 05A18, 16T30 |
| url | https://arxiv.org/abs/2403.19744 |