"Kazhdan-Lusztig conjecture"의 두 판 사이의 차이
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| + | ==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 | ||
| + | |||
| + | |||
| + | ==related items== | ||
| + | * [[BGG category]] | ||
| + | * [[Hecke algebra]] | ||
| + | * [[Enumerative problems and Schubert calculus]] | ||
| + | * [[Flag manifold and flag variety]] | ||
| + | |||
| + | |||
| + | ==exposition== | ||
| + | * [https://docs.google.com/file/d/0B8XXo8Tve1cxd2JGOUFfSG5nbjQ/edit Williamson- Kazhdan-Lusztig conjecture and shadows of Hodge theory] | ||
| + | |||
| + | |||
| + | ==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. | ||
| + | |||
| + | [[분류:Hecke algebra]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 14일 (토) 00:56 판
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.