Schur Rank, Compatibility Degree, and Canonical Decomposition
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909636091707392 |
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| author | Fei, Jiarui |
| author_facet | Fei, Jiarui |
| contents | The notion of denominator vectors can be extended to all generic basis elements of upper cluster algebras in a natural way. Under a weakened version of generic pairing assumption, we provide a representation-theoretic interpretation for this extended notion. We derive several consequences in this generality. We present a counterexample to the conjecture that distinct cluster monomials have distinct denominator vectors. Utilizing a new rank function called the Schur rank, we extend the notion of compatibility degree. As an application, we find a tropical method to compute the multiplicity of a real component in the canonical decomposition of $δ$-vectors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schur Rank, Compatibility Degree, and Canonical Decomposition Fei, Jiarui Representation Theory Combinatorics Rings and Algebras Primary 13F60, 16G10, Secondary 14T99 The notion of denominator vectors can be extended to all generic basis elements of upper cluster algebras in a natural way. Under a weakened version of generic pairing assumption, we provide a representation-theoretic interpretation for this extended notion. We derive several consequences in this generality. We present a counterexample to the conjecture that distinct cluster monomials have distinct denominator vectors. Utilizing a new rank function called the Schur rank, we extend the notion of compatibility degree. As an application, we find a tropical method to compute the multiplicity of a real component in the canonical decomposition of $δ$-vectors. |
| title | Schur Rank, Compatibility Degree, and Canonical Decomposition |
| topic | Representation Theory Combinatorics Rings and Algebras Primary 13F60, 16G10, Secondary 14T99 |
| url | https://arxiv.org/abs/2503.12700 |