Differential Algebraic Closure Framework for Group Representation Theory: Explicit Solutions and Combinatorial Structures
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| Natura: | Recurso digital |
| Lingua: | inglese |
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Zenodo
2025
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| _version_ | 1866901124657709056 |
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| author | liu, shifa |
| author_facet | liu, shifa |
| contents | <p>This paper extends the differential algebraic closure framework from polynomial equations and partial differential equations to the realm of group representation theory. We construct a rigorously defined group representation closure KG and demonstrate that irreducible character decompositions and representation matrices for finite groups and compact Lie groups can be explicitly expressed within this closure.We provide a detailed constructive framework with complete proofs, derive corrected combinatorial expressions with rigorous connections to symmetric group representations, present a complete algorithmic<br>description with complexity analysis, and situate our results within classical representation theory and differential Galois theory. Numerical experiments across diverse group families confirm spectral conver gence and demonstrate the necessity of combinatorial corrections for higher-dimensional representations. This work establishes that explicit analytic expressions exist in the appropriately constructed group representation closure KG for a significant class of group representation problems, providing a new algebraic perspective on representation-theoretic solvability while maintaining consistency with classical<br>complexity results.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18094806 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Differential Algebraic Closure Framework for Group Representation Theory: Explicit Solutions and Combinatorial Structures liu, shifa Group representations, Differential algebraic closure, Character theory, Explicit solu tions, Combinatorial group theory, Computational algebra, Symmetric groups, Representation varieties, Schur-Weyl duality <p>This paper extends the differential algebraic closure framework from polynomial equations and partial differential equations to the realm of group representation theory. We construct a rigorously defined group representation closure KG and demonstrate that irreducible character decompositions and representation matrices for finite groups and compact Lie groups can be explicitly expressed within this closure.We provide a detailed constructive framework with complete proofs, derive corrected combinatorial expressions with rigorous connections to symmetric group representations, present a complete algorithmic<br>description with complexity analysis, and situate our results within classical representation theory and differential Galois theory. Numerical experiments across diverse group families confirm spectral conver gence and demonstrate the necessity of combinatorial corrections for higher-dimensional representations. This work establishes that explicit analytic expressions exist in the appropriately constructed group representation closure KG for a significant class of group representation problems, providing a new algebraic perspective on representation-theoretic solvability while maintaining consistency with classical<br>complexity results.</p> |
| title | Differential Algebraic Closure Framework for Group Representation Theory: Explicit Solutions and Combinatorial Structures |
| topic | Group representations, Differential algebraic closure, Character theory, Explicit solu tions, Combinatorial group theory, Computational algebra, Symmetric groups, Representation varieties, Schur-Weyl duality |
| url | https://doi.org/10.5281/zenodo.18094806 |