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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.05369 |
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| _version_ | 1866909985490862080 |
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| author | Petroulakis, George |
| author_facet | Petroulakis, George |
| contents | This article introduces a framework for the localization and isolation of singularities in the affine Grassmannian. Our primary result is a structural factorization of the transition matrix $C$ between the Mirković--Vilonen (MV) basis and the convolution basis into $C = P \cdot M \cdot A \cdot Q^{-1}$, where the four factors represent: equivariant localization ($Q$), fusion via nearby cycles ($A$), local intersection cohomology stalks ($M$), and diagonal normalization ($P$). Utilizing this factorization, and by introducing the Geometric Efficiency metric ($η$) we establish a Universal Geometric Rank Bound, proving that the rank of the transition matrix $C$ is bounded by the dimension of the local Braden--MacPherson (BMP) stalks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05369 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Localization of Singularities and Universal Geometric Rank Bounds in the Satake Correspondence Petroulakis, George Algebraic Geometry Representation Theory This article introduces a framework for the localization and isolation of singularities in the affine Grassmannian. Our primary result is a structural factorization of the transition matrix $C$ between the Mirković--Vilonen (MV) basis and the convolution basis into $C = P \cdot M \cdot A \cdot Q^{-1}$, where the four factors represent: equivariant localization ($Q$), fusion via nearby cycles ($A$), local intersection cohomology stalks ($M$), and diagonal normalization ($P$). Utilizing this factorization, and by introducing the Geometric Efficiency metric ($η$) we establish a Universal Geometric Rank Bound, proving that the rank of the transition matrix $C$ is bounded by the dimension of the local Braden--MacPherson (BMP) stalks. |
| title | Localization of Singularities and Universal Geometric Rank Bounds in the Satake Correspondence |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2601.05369 |