"Kazhdan-Lusztig conjecture"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
||
| 1번째 줄: | 1번째 줄: | ||
==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 | ||
| 10번째 줄: | 10번째 줄: | ||
==Hecke algebra== | ==Hecke algebra== | ||
| − | * basis of Hecke algebra | + | * basis of Hecke algebra <math>\{H_{x}| x\in W\}</math> |
| − | * new basis of Hecke algebra | + | * 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== | ||
| 36번째 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. | ||
2020년 11월 16일 (월) 10:07 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.