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Combinatorics of highest weight crystals of type D : D형 최고 무게 결정의 조합론
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | 권재훈 | - |
dc.contributor.author | 장일승 | - |
dc.date.accessioned | 2021-11-30T04:50:05Z | - |
dc.date.available | 2021-11-30T04:50:05Z | - |
dc.date.issued | 2021-02 | - |
dc.identifier.other | 000000165166 | - |
dc.identifier.uri | https://hdl.handle.net/10371/176024 | - |
dc.identifier.uri | https://dcollection.snu.ac.kr/common/orgView/000000165166 | ko_KR |
dc.description | 학위논문 (박사) -- 서울대학교 대학원 : 자연과학대학 수리과학부, 2021. 2. 권재훈. | - |
dc.description.abstract | In this thesis, we study the crystals of type D from a combinatorial viewpoint. We focus on especially the crystals $B(\lambda)$ and $B(\infty)$, where $B(\infty)$ is the crystal of the negative half of the quantum group and $B(\lambda)$ is the crystal of an integrable highest weight irreducible module with highest weight $\lambda$.
As a main result, we obtain a simple description of the crystal structure of $B(\infty)$ in terms of Lusztig's parametrization using the PBW basis associated with a certain reduced expression of the longest element of the Weyl group. Also, we develop a combinatorial algorithm on $B(\lambda)$, which is compatible with the crystal structure of $B(\infty)$. These results establish an explicit combinatorial description of the crystal embedding from $B(\lambda)$ into $B(\infty)$. Our study of the crystal structure of $B(\lambda)$ and $B(\infty)$ has several interesting applications such as an affine crystal theoretic interpretation of Robinson-Schensted-Knuth type correspondence of type D, a new formula for the branching multiplicity from ${\rm GL}_n$ to ${\rm O}_n$, and a new combinatorial model of Kirillov-Reshetikhin crystals of type $\text{D}_n^{(1)}$ associated with the spin node. | - |
dc.description.abstract | 본 학위논문에서는 조합론적인 관점에서 D형 결정을 연구한다. 특히, 양자 군의 음의 부분의 결정 $B(\infty)$과 최고 무게가 $\lambda$인 가적 최고 무게 기약 모듈의 결정 $B(\lambda)$을 중점적으로 연구한다.
본 학위논문의 주요 결과로써, PBW기저에 의한 루스티그의 매개화를 이용하여 $B(\infty)$의 결정 구조를 명확하게 제시하고, $B(\lambda)$에서 PBW기저의 결정 구조와 양립하는 조합론적 알고리즘을 개발한다. 그리고 이러한 결과로부터 $B(\lambda)$에서 $B(\infty)$으로의 결정 매입의 조합론적 모형을 얻는다. 위 $B(\lambda)$와 $B(\infty)$의 결정 구조 연구의 응용으로 D형 로빈슨-셴스티드-커누스 대응의 아핀 결정 이론적 해석, ${\rm GL}_n$에서 ${\rm O}_n$으로의 분지 중복도에 대한 새로운 조합론적 공식 그리고 스핀점과 연관된 D형 키릴로프-레셰티킨 결정의 조합론적 모형을 얻는다. | - |
dc.description.tableofcontents | Abstract i
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 RSK correspondence of type D . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Branching rules for $({\rm GL}_n, {\rm O}_n)$ . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Kirillov-Reshetikhin crystals of type $\text{D}_n^{(1)}$ associated with spin node 6 1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Crystal bases 9 2.1 Quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Representations of quantum groups . . . . . . . . . . . . . . . . . . 11 2.1.2 Crystal bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Crystal base of $U_q^−(\mathfrak{g})$ . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 PBW basis and crystals . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.3 Quantum nilpotent subalgebras . . . . . . . . . . . . . . . . . . . . 26 3 PBW crystal and RSK correspondence of type D 28 3.1 Robinson-Schensted-Knuth correspondence . . . . . . . . . . . . . . . . . . 28 3.1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 Crystals and RSK correspondence . . . . . . . . . . . . . . . . . . . 30 3.2 PBW crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.1 Description of $\tilde{f}_i$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.2 Kac-Moody algebra of type $\text{D}_n$ . . . . . . . . . . . . . . . . . . . . 35 3.2.3 PBW crystal of type $\text{D}_n$ . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.4 Crystal ${\bf B}^J$ of quantum nilpotent subalgebra . . . . . . . . . . . . . 41 3.2.5 Notation for ${\bf B}^J$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Burge correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 RSK of type $\text{D}_n$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 Shape formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Crystal embedding from $B(\lambda)$ into $B(\infty)$ . . . . . . 50 4.1 Highest weight crystals for type $\text{D}_n$ . . . . . . . . . . . . . . . . . . . . . . 51 4.1.1 Tableaux with two columns . . . . . . . . . . . . . . . . . . . . . . 51 4.1.2 Kashiwara-Nakashima tableaux of type $\text{D}_n$ . . . . . . . . . . . . . . 52 4.1.3 Spinor model for type $\text{D}_n$ . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Isomorphism from ${\bf KN}_{\lambda}$ to ${\bf T}_{\lambda}$ . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Separation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.1 Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.2 Separation when $\lambda \ge 0$ . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.3 Separation when $\lambda_n < 0$ . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4 Embedding from ${\bf T}_{\lambda}$ into ${\bf V}_{\lambda}$ . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.1 Crystal of parabolic Verma module . . . . . . . . . . . . . . . . . . 72 4.4.2 Embedding as $\mathfrak{g}$-crystals . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Lusztig data of Kashiwara-Nakashima tableaux of type D . . . . . . . . . . 80 5 Branching rules for classical groups 84 5.1 Littlewood-Richardson tableaux . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Howe duality on Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.1 Kac-Moody algebra of type $\text{D}_n^{(1)}$ . . . . . . . . . . . . . . . . . . . . 87 5.2.2 Dual pair $({\rm O}_n, \mathfrak{d}_{\infty})$ on Fock space . . . . . . . . . . . . . . . . . . . 88 5.3 Separation on l-highest weight vectors . . . . . . . . . . . . . . . . . . . . . 89 5.3.1 Revisit of spinor model over $U_q(D_{\infty})$ . . . . . . . . . . . . . . . . . 90 5.3.2 $\mathfrak{l}$-highest weight vectors . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.3 Sliding on $\mathfrak{l}$-highest weight vectors . . . . . . . . . . . . . . . . . . . 98 5.3.4 Separation on $\mathfrak{l}$-highest weight vectors . . . . . . . . . . . . . . . . 101 5.4 Branching rules from $\text{D}_{\infty}$ to $\text{A}_{+\infty}$ . . . . . . . . . . . . . . . . . . . . . . . 104 5.4.1 Branching multiplicity formulas from $\text{D}_{\infty}$ to $\text{A}_{+\infty}$ . . . . . . . . . . 104 5.4.2 Branching multiplicity formulas from ${\rm GL}_n$ to ${\rm O}_n$ . . . . . . . . . . . 110 5.4.3 Comparing other works . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5 Generalized exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.1 Generalized exponents . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.2 Combinatorial formula of generalized exponents . . . . . . . . . . . 115 6 Affine crystals 120 6.1 Quantum affine algebras and crystals . . . . . . . . . . . . . . . . . . . . . 120 6.2 Kirillov-Reshetikhin crystals $B^{n, s}$ of type $\text{D}_n^{(1)}$ . . . . . . . . . . . . . . . . 121 6.3 Burge correspondence of type $\text{D}_n^{(1)}$ . . . . . . . . . . . . . . . . . . . . . . . 124 7 Proofs 128 7.1 In Chapters 3 and 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.1.1 Formula of Berenstein-Zelevinsky . . . . . . . . . . . . . . . . . . . 128 7.1.2 Proof of Theorem 3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.1.3 Proofs of Theorems 3.3.6 and 6.2.4 . . . . . . . . . . . . . . . . . . 134 7.2 In Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.1 Proof of Lemma 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.2 Proof of Lemma 4.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3 In Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.3.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.3.2 Proof of Theorem 5.4.4 when when $n − 2\mu_1' \ge 0$ . . . . . . . . . . . 151 7.3.3 Proof of Theorem 5.4.4 when when $n − 2\mu_1' < 0$ . . . . . . . . . . . 161 Appendices 164 A Index of notation, Table and Figure 165 A.1 Index of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A.1.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A.1.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A.1.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.1.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.1.5 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.2 Crystal graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 A.3 Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Abstract (in Korean) 176 Acknowledgement (in Korean) 177 | - |
dc.format.extent | v, 175 | - |
dc.language.iso | eng | - |
dc.publisher | 서울대학교 대학원 | - |
dc.subject | Quantum groups | - |
dc.subject | Crystal bases | - |
dc.subject | Kirillov-Reshetikhin crystals | - |
dc.subject | Robinson-Schensted-Knuth correspondence | - |
dc.subject | Branching rules | - |
dc.subject | Generalized exponents | - |
dc.subject | 양자군 | - |
dc.subject | 결정 기저 | - |
dc.subject | 키릴로프-레셰티킨 결정 | - |
dc.subject | 로빈슨-셴스테드-커누스 대응 | - |
dc.subject | 분지 규칙 | - |
dc.subject | 일반화된 지수 | - |
dc.subject.ddc | 510 | - |
dc.title | Combinatorics of highest weight crystals of type D | - |
dc.title.alternative | D형 최고 무게 결정의 조합론 | - |
dc.type | Thesis | - |
dc.type | Dissertation | - |
dc.contributor.AlternativeAuthor | Il-Seung Jang | - |
dc.contributor.department | 자연과학대학 수리과학부 | - |
dc.description.degree | Doctor | - |
dc.date.awarded | 2021-02 | - |
dc.identifier.uci | I804:11032-000000165166 | - |
dc.identifier.holdings | 000000000044▲000000000050▲000000165166▲ | - |
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