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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.10069 |
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| _version_ | 1866910186383343616 |
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| author | Ikenmeyer, Christian Omar, Heidi Tsintsilidas, Dimitrios |
| author_facet | Ikenmeyer, Christian Omar, Heidi Tsintsilidas, Dimitrios |
| contents | We construct an explicit field-independent SL$_2$-equivariant isomorphism between an invariant space of tensors and a plethysm space. The existence of such an isomorphism was only known in characteristic 0, and only indirectly via character theory. Our isomorphism naturally extends the web of field-independent isomorphisms given by Hermite reciprocity, Hodge duality, and the Wronskian isomorphism. This is a characteristic free generalization of a classical situation in characteristic zero: certain rectangular Kronecker coefficients coincide with certain plethysm coefficients, and their non-negativiy proves the unimodality of the $q$-binomial coefficient.
We also give a short combinatorial field-independent proof that the Hermite reciprocity map over the standard basis is a triangular matrix with 1s on the main diagonal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Field-independent Kronecker-plethysm isomorphisms Ikenmeyer, Christian Omar, Heidi Tsintsilidas, Dimitrios Representation Theory Combinatorics 20G15 (Primary) 05E15 (Secondary) We construct an explicit field-independent SL$_2$-equivariant isomorphism between an invariant space of tensors and a plethysm space. The existence of such an isomorphism was only known in characteristic 0, and only indirectly via character theory. Our isomorphism naturally extends the web of field-independent isomorphisms given by Hermite reciprocity, Hodge duality, and the Wronskian isomorphism. This is a characteristic free generalization of a classical situation in characteristic zero: certain rectangular Kronecker coefficients coincide with certain plethysm coefficients, and their non-negativiy proves the unimodality of the $q$-binomial coefficient. We also give a short combinatorial field-independent proof that the Hermite reciprocity map over the standard basis is a triangular matrix with 1s on the main diagonal. |
| title | Field-independent Kronecker-plethysm isomorphisms |
| topic | Representation Theory Combinatorics 20G15 (Primary) 05E15 (Secondary) |
| url | https://arxiv.org/abs/2509.10069 |