"Kazhdan-Lusztig conjecture"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 6개는 보이지 않습니다) | |||
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==introduction== | ==introduction== | ||
| − | * The Kazhdan-Lusztig theory provides the solution to the problem of determining the irreducible characters in the BGG category | + | * The Kazhdan-Lusztig theory provides the solution to the problem of determining the irreducible characters in the BGG category <math>\mathcal{O}</math> of semisimple Lie algebras ([KL], [BB], [BK]). |
* The theory was originally formulated in terms of the canonical bases (i.e., Kazhdan-Lusztig bases) of Hecke algebras. | * The theory was originally formulated in terms of the canonical bases (i.e., Kazhdan-Lusztig bases) of Hecke algebras. | ||
* 1979 conjectures | * 1979 conjectures | ||
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==Hecke algebra== | ==Hecke algebra== | ||
| − | * new basis of Hecke algebra | + | * basis of Hecke algebra <math>\{H_{x}| x\in W\}</math> |
| − | + | * new basis of Hecke algebra <math>\{\underline{H}_{x}| x\in W\}</math> | |
| + | :<math> | ||
\underline{H}_{x}=H_{x}+\sum_{y\in W, \ell(y)<\ell(x)} h_{y,x}H_{y} | \underline{H}_{x}=H_{x}+\sum_{y\in W, \ell(y)<\ell(x)} h_{y,x}H_{y} | ||
| − | + | </math> | |
| − | where | + | where <math>h_{y,x}\in v\mathbb{Z}[v]</math> is so called the Kazhdan-Lusztig polynomial |
| − | * positivity conjecture : | + | * positivity conjecture : <math>h_{x,y}\in \mathbb{Z}_{\geq 0}[v]</math> |
| − | |||
| − | |||
==Hodge theory== | ==Hodge theory== | ||
| 37번째 줄: | 36번째 줄: | ||
==articles== | ==articles== | ||
| − | * [BB] A. Beilinson and J. Bernstein, Localisation de | + | * [BB] A. Beilinson and J. Bernstein, Localisation de <math>\mathfrak g</math>-modules, C.R. Acad. Sci. Paris Ser. I Math. 292 (1981), 15-18. |
* [BK] J.L.Brylinski and M.Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. | * [BK] J.L.Brylinski and M.Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. | ||
* [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. | * [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. | ||
[[분류:Hecke algebra]] | [[분류:Hecke algebra]] | ||
| + | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q6381065 Q6381065] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'kazhdan'}, {'OP': '*'}, {'LOWER': 'lusztig'}, {'LEMMA': 'polynomial'}] | ||
2021년 2월 17일 (수) 01:07 기준 최신판
introduction
- The Kazhdan-Lusztig theory provides the solution to the problem of determining the irreducible characters in the BGG category \(\mathcal{O}\) of semisimple Lie algebras ([KL], [BB], [BK]).
- The theory was originally formulated in terms of the canonical bases (i.e., Kazhdan-Lusztig bases) of Hecke algebras.
- 1979 conjectures
- KL character formula
- KL positivity conjecture
- Kazhdan-Lusztig polynomial
Hecke algebra
- basis of Hecke algebra \(\{H_{x}| x\in W\}\)
- new basis of Hecke algebra \(\{\underline{H}_{x}| x\in W\}\)
\[ \underline{H}_{x}=H_{x}+\sum_{y\in W, \ell(y)<\ell(x)} h_{y,x}H_{y} \] where \(h_{y,x}\in v\mathbb{Z}[v]\) is so called the Kazhdan-Lusztig polynomial
- positivity conjecture \[h_{x,y}\in \mathbb{Z}_{\geq 0}[v]\]
Hodge theory
- Poincare duality
- hard Lefshetz theorem
- Hodge-Riemann bilinear relation
- BGG category
- Hecke algebra
- Enumerative problems and Schubert calculus
- Flag manifold and flag variety
exposition
articles
- [BB] A. Beilinson and J. Bernstein, Localisation de \(\mathfrak g\)-modules, C.R. Acad. Sci. Paris Ser. I Math. 292 (1981), 15-18.
- [BK] J.L.Brylinski and M.Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410.
- [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.
메타데이터
위키데이터
- ID : Q6381065
Spacy 패턴 목록
- [{'LOWER': 'kazhdan'}, {'OP': '*'}, {'LOWER': 'lusztig'}, {'LEMMA': 'polynomial'}]