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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.23286 |
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| _version_ | 1866915958411493376 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{λ\, μ\, ν}$. A notable advance in this direction is due to Blasiak, who proved a combinatorial interpretation in terms of colored Yamanouchi tableaux for the case whereby one of the indexing partitions is hook-shaped. In this paper, we introduce a framework for the evaluation and combinatorial interpretation of Kronecker coefficients, combining a Schur function identity of Littlewood, the Giambelli identity for Schur functions, and Blasiak's combinatorial rule. This framework reduces the study of Kronecker coefficients to alternating sums involving hook-indexed cases. As an application of this framework, we obtain combinatorial interpretations of $g_{t, h^{(1)}, h^{(2)}}$ for two-row partitions $t$ and hook-like partitions $h^{(1)}$ and $h^{(2)}$ satisfying natural conditions. More broadly, our approach provides a systematic method for extending hook-based combinatorial rules to wider families of Kronecker coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23286 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kronecker coefficients via the Giambelli identity for Schur functions Campbell, John M. Combinatorics 05E10, 20C30 One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{λ\, μ\, ν}$. A notable advance in this direction is due to Blasiak, who proved a combinatorial interpretation in terms of colored Yamanouchi tableaux for the case whereby one of the indexing partitions is hook-shaped. In this paper, we introduce a framework for the evaluation and combinatorial interpretation of Kronecker coefficients, combining a Schur function identity of Littlewood, the Giambelli identity for Schur functions, and Blasiak's combinatorial rule. This framework reduces the study of Kronecker coefficients to alternating sums involving hook-indexed cases. As an application of this framework, we obtain combinatorial interpretations of $g_{t, h^{(1)}, h^{(2)}}$ for two-row partitions $t$ and hook-like partitions $h^{(1)}$ and $h^{(2)}$ satisfying natural conditions. More broadly, our approach provides a systematic method for extending hook-based combinatorial rules to wider families of Kronecker coefficients. |
| title | Kronecker coefficients via the Giambelli identity for Schur functions |
| topic | Combinatorics 05E10, 20C30 |
| url | https://arxiv.org/abs/2604.23286 |