Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests
We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov...
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| Format: | Article |
| Language: | English |
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Springer Science and Business Media LLC
2024
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| Online Access: | https://hdl.handle.net/1721.1/155277 |
| _version_ | 1811082295060201472 |
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| author | Gavrilova, Svetlana Petrov, Leonid |
| author_facet | Gavrilova, Svetlana Petrov, Leonid |
| author_sort | Gavrilova, Svetlana |
| collection | MIT |
| description | We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4×4 problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size 𝑛≥, which appear new for 𝑛≥5. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation. |
| first_indexed | 2024-09-23T12:00:29Z |
| format | Article |
| id | mit-1721.1/155277 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:00:29Z |
| publishDate | 2024 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1552772024-09-19T05:26:20Z Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests Gavrilova, Svetlana Petrov, Leonid We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4×4 problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size 𝑛≥, which appear new for 𝑛≥5. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation. 2024-06-17T15:37:02Z 2024-06-17T15:37:02Z 2024-06-10 2024-06-16T03:13:02Z Article http://purl.org/eprint/type/JournalArticle 1022-1824 1420-9020 https://hdl.handle.net/1721.1/155277 Gavrilova, S., Petrov, L. Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests. Sel. Math. New Ser. 30, 56 (2024). PUBLISHER_CC en 10.1007/s00029-024-00945-3 Selecta Mathematica Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Science and Business Media LLC Springer International Publishing |
| spellingShingle | Gavrilova, Svetlana Petrov, Leonid Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests |
| title | Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests |
| title_full | Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests |
| title_fullStr | Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests |
| title_full_unstemmed | Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests |
| title_short | Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests |
| title_sort | tilted biorthogonal ensembles grothendieck random partitions and determinantal tests |
| url | https://hdl.handle.net/1721.1/155277 |
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