Temperley-Lieb Crystals
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917600650330112 |
|---|---|
| author | Nguyen, Son Pylyavskyy, Pavlo |
| author_facet | Nguyen, Son Pylyavskyy, Pavlo |
| contents | Elements of Lusztig's dual canonical bases are Schur-positive when evaluated on (generalized) Jacobi-Trudi matrices. This deep property was proved by Rhoades and Skandera, relying on a result of Haiman, and ultimately on the (proof of) Kazhdan-Lusztig conjecture. For a particularly tractable part of the dual canonical basis - called Temperley-Lieb immanants - we give a generalization of Littlewood-Richardson rule: we provide a combinatorial interpretation for the coefficient of a particular Schur function in the evaluation of a particular Temperley-Lieb immanant on a particular Jacobi-Trudi matrix. For this we introduce shuffle tableaux, and apply Stembridge's axioms to show that certain graphs on shuffle tableaux are type $A$ Kashiwara crystals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18716 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Temperley-Lieb Crystals Nguyen, Son Pylyavskyy, Pavlo Combinatorics Quantum Algebra Rings and Algebras Representation Theory Elements of Lusztig's dual canonical bases are Schur-positive when evaluated on (generalized) Jacobi-Trudi matrices. This deep property was proved by Rhoades and Skandera, relying on a result of Haiman, and ultimately on the (proof of) Kazhdan-Lusztig conjecture. For a particularly tractable part of the dual canonical basis - called Temperley-Lieb immanants - we give a generalization of Littlewood-Richardson rule: we provide a combinatorial interpretation for the coefficient of a particular Schur function in the evaluation of a particular Temperley-Lieb immanant on a particular Jacobi-Trudi matrix. For this we introduce shuffle tableaux, and apply Stembridge's axioms to show that certain graphs on shuffle tableaux are type $A$ Kashiwara crystals. |
| title | Temperley-Lieb Crystals |
| topic | Combinatorics Quantum Algebra Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2402.18716 |