"Kazhdan-Lusztig conjecture"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
 
(사용자 2명의 중간 판 4개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 +
==introduction==
 +
* 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.
 +
* 1979 conjectures
 +
** KL character formula
 +
** KL positivity conjecture
 +
* [[Kazhdan-Lusztig polynomial]]
  
 +
 +
 +
==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}
 +
</math>
 +
where <math>h_{y,x}\in v\mathbb{Z}[v]</math> is so called the Kazhdan-Lusztig polynomial
 +
* positivity conjecture : <math>h_{x,y}\in \mathbb{Z}_{\geq 0}[v]</math>
 +
 +
==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 <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.
 +
* [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.
 +
 +
[[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


related items


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

Spacy 패턴 목록

  • [{'LOWER': 'kazhdan'}, {'OP': '*'}, {'LOWER': 'lusztig'}, {'LEMMA': 'polynomial'}]