Discovering product and coproduct rules for bases of QSym_F through supercharacters
- 주제어 (키워드) quasisymmetric function , Hopf algebra , Hall-Littlewood function , shuffle , supercharacter , categorification
- 발행기관 서강대학교 일반대학원
- 지도교수 오영탁
- 발행년도 2024
- 학위수여년월 2024. 2
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/sogang/000000076651
- UCI I804:11029-000000076651
- 본문언어 영어
- 저작권 서강대학교 논문은 저작권 보호를 받습니다.
초록
In this paper, we establish product and coproduct rules for three bases of the Hopf algebra QSym_F of quasisymmetric functions over F, with F being either C(q, t) or C(q). These results are derived through the categorizations of QSym_C obtained by utilizing the normal lattice supercharacter theories. Firstly, we deal with a basis {D_α(q, t) | α ∈ Comp} of QSym_C(q,t), where Comp denotes the set of all compositions. This basis is obtained from the direct sum of specific supercharacter function spaces and consists of superclass identifier functions. Upon appropriate specializations of q and t, it yields notable bases of QSym_C and QSym_C(q), including enriched q-monomial quasisymmetric functions introduced by Grinberg and Vassilieva. Secondly, we deal with the basis {G_α(q) | α ∈ Comp} of QSym_C(q), where G_α(q) represents the quasisymmetric Hall–Littlewood function introduced by Hivert. Our product rule is new, whereas our coproduct rule turns out to be equivalent to the existing coproduct rule of Hivert. Finally, we consider a basis {M_α(q) | α ∈ Comp} of QSym_C(q), where M_α(q) is a q-analogue of the monomial quasisymmetric function. Key words: quasisymmetric function, Hopf algebra, Hall–Littlewood function, shuffle, supercharacter, categorification
more목차
1 Introduction 4
2 Preliminaries 8
2.1 The Hopf algebras in our consideration 8
2.1.1 The Hopf algebra of quasisymmetric functions 9
2.1.2 The Hopf algebra of noncommutative symmetric functions 10
2.1.3 The Hopf algebra of free quasisymmetric functions 11
2.2 Supercharacter theories of a finite group 13
3 Categorifications of QSym using supercharacter theories 17
3.1 Normal lattice supercharacter theories of ⊕C_ν 17
3.2 A Hopf algebra structure of ⊕n≥0 scf(Nn(ν)) 19
3.3 The superclass identifiers of ⊕n≥0 scf(Nn(ν)) 31
4 A new basis {D_α(q, t) | α ∈ Comp} of QSym_C(q,t) 37
4.1 The definition and notable properties of D_α(q, t) 37
4.2 Product and coproduct rules for {D_α(q, t)} 42
4.3 The proof of Theorem 4.8 45
4.4 Antipode formula for {D_α(q, t)} 55
5 Quasisymmetric Hall–Littlewood functions 58
5.1 Hivert’s quasisymmetric Hall–Littlewood functions 58
5.2 Class functions corresponding to quasisymmetric Hall–Littlewood functions 62
5.3 Product and coproduct formulas for {G_I(q)} 64
5.4 The proofs of Theorem 5.6 and Theorem 5.9 67
6 A q analogue of monomial quasisymmetric function 74
Bibliography 77
