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Tilted Richardson Varieties
Tilted Richardson Varieties
Detailed Information
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202103552
- ISBN
- 9798280718753
- DDC
- 510
- 저자명
- Gao, Jiyang.
- 서명/저자
- Tilted Richardson Varieties
- 발행사항
- [Sl] : Harvard University, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 156 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
- 주기사항
- Advisor: Williams, Lauren.
- 학위논문주기
- Thesis (Ph.D.)--Harvard University, 2025.
- 초록/해제
- 요약The flag variety Fln, and its subvarieties, such as Schubert and Richardson varieties, are central objects in algebraic geometry and algebraic combinatorics. In this thesis, we intro-duce and study tilted Richardson varieties Tu,v, a new family of subvarieties of Fln, defined for all pairs of permutations u and v. This family generalizes classical Richardson varieties when u ≤ v in the Bruhat order and provides a geometric framework for the quantum Bruhat graph. We establish their fundamental geometric properties, including irreducibility, explicit dimension formulas, and a well-defined stratification indexed by tilted Bruhat intervals, a generalization of classical Bruhat intervals introduced by Brenti, Fomin, and Postnikov. Moreover, we introduce a tilted generalization of the classical Deodhar decom-position, which leads to a combinatorial formula for tilted Kazhdan-Lusztig R-polynomials, a notion that arises naturally in our framework. We further develop a theory of total positivity for tilted Richardson varieties. In par-ticular, we define and study their totally nonnegative parts and prove that they form CW-complexes. This generalizes previous work on the totally nonnegative flag variety and ad-dresses Bjorner's questions regarding geometric realizations of tilted Bruhat intervals. Finally, we establish explicit connections between tilted Richardson varieties and quantum Schubert calculus. We prove that Tu,v coincides with minimal-degree two-point curve neighborhoods, allowing us to compute their cohomology classes and derive new relationships among Gromov-Witten invariants of the flag variety.
- 일반주제명
- Mathematics
- 일반주제명
- Theoretical mathematics
- 키워드
- Symmetric group
- 키워드
- Bruhat intervals
- 키워드
- Flag variety
- 기타저자
- Harvard University Mathematics
- 기본자료저록
- Dissertations Abstracts International. 86-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■006m o d
■007cr#unu||||||||
■020 ▼a9798280718753
■035 ▼a(MiAaPQ)AAI32041912
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aGao, Jiyang.▼0(orcid)0000-0003-0703-7183
■24510▼aTilted Richardson Varieties
■260 ▼a[Sl]▼bHarvard University▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a156 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 86-12, Section: B.
■500 ▼aAdvisor: Williams, Lauren.
■5021 ▼aThesis (Ph.D.)--Harvard University, 2025.
■520 ▼aThe flag variety Fln, and its subvarieties, such as Schubert and Richardson varieties, are central objects in algebraic geometry and algebraic combinatorics. In this thesis, we intro-duce and study tilted Richardson varieties Tu,v, a new family of subvarieties of Fln, defined for all pairs of permutations u and v. This family generalizes classical Richardson varieties when u ≤ v in the Bruhat order and provides a geometric framework for the quantum Bruhat graph. We establish their fundamental geometric properties, including irreducibility, explicit dimension formulas, and a well-defined stratification indexed by tilted Bruhat intervals, a generalization of classical Bruhat intervals introduced by Brenti, Fomin, and Postnikov. Moreover, we introduce a tilted generalization of the classical Deodhar decom-position, which leads to a combinatorial formula for tilted Kazhdan-Lusztig R-polynomials, a notion that arises naturally in our framework. We further develop a theory of total positivity for tilted Richardson varieties. In par-ticular, we define and study their totally nonnegative parts and prove that they form CW-complexes. This generalizes previous work on the totally nonnegative flag variety and ad-dresses Bjorner's questions regarding geometric realizations of tilted Bruhat intervals. Finally, we establish explicit connections between tilted Richardson varieties and quantum Schubert calculus. We prove that Tu,v coincides with minimal-degree two-point curve neighborhoods, allowing us to compute their cohomology classes and derive new relationships among Gromov-Witten invariants of the flag variety.
■590 ▼aSchool code: 0084.
■650 4▼aMathematics
■650 4▼aTheoretical mathematics
■653 ▼aSymmetric group
■653 ▼aRichardson varieties
■653 ▼aAlgebraic combinatorics
■653 ▼aBruhat intervals
■653 ▼aFlag variety
■690 ▼a0405
■690 ▼a0642
■71020▼aHarvard University▼bMathematics.
■7730 ▼tDissertations Abstracts International▼g86-12B.
■790 ▼a0084
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17357732▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
