Maximal Newton polygons via the quantum Bruhat graph
This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bru...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2012-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3092/pdf |
| _version_ | 1797270322886475776 |
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| author | Elizabeth T. Beazley |
| author_facet | Elizabeth T. Beazley |
| author_sort | Elizabeth T. Beazley |
| collection | DOAJ |
| description | This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety. |
| first_indexed | 2024-04-25T02:02:26Z |
| format | Article |
| id | doaj.art-fb3c0cd5f4704408b2cefc5dc0f2e1dc |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:26Z |
| publishDate | 2012-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-fb3c0cd5f4704408b2cefc5dc0f2e1dc2024-03-07T14:51:45ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502012-01-01DMTCS Proceedings vol. AR,...Proceedings10.46298/dmtcs.30923092Maximal Newton polygons via the quantum Bruhat graphElizabeth T. Beazley0Department of Mathematics & Statistics [Williamstown]This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety.https://dmtcs.episciences.org/3092/pdfquantum bruhat graphquantum schubert calculusaffine weyl groupnewton polygonaffine deligne-lusztig varietymazur's inequality[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Elizabeth T. Beazley Maximal Newton polygons via the quantum Bruhat graph Discrete Mathematics & Theoretical Computer Science quantum bruhat graph quantum schubert calculus affine weyl group newton polygon affine deligne-lusztig variety mazur's inequality [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Maximal Newton polygons via the quantum Bruhat graph |
| title_full | Maximal Newton polygons via the quantum Bruhat graph |
| title_fullStr | Maximal Newton polygons via the quantum Bruhat graph |
| title_full_unstemmed | Maximal Newton polygons via the quantum Bruhat graph |
| title_short | Maximal Newton polygons via the quantum Bruhat graph |
| title_sort | maximal newton polygons via the quantum bruhat graph |
| topic | quantum bruhat graph quantum schubert calculus affine weyl group newton polygon affine deligne-lusztig variety mazur's inequality [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/3092/pdf |
| work_keys_str_mv | AT elizabethtbeazley maximalnewtonpolygonsviathequantumbruhatgraph |