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Algebra, Combination and Number Theory

Time: Mar 24 , 2023
Algebra, Combination and Number Theory


Algebra research at Peking University has a long tradition and covers a wide range of topics across group and representation theory, algebraic number theory, Lie algebras and algebraic combinatorics. These areas lie at the heart of mathematics.
Group and representation theory gives a mathematical abstraction for the study of symmetry and thus have found profound applications across the physical sciences, for example in the chemistry of molecular vibrations and in particle physics, and have a close relationship with geometry, number theory and combinatorics. The research interests of group and representation theory in Peking University involves permutation groups, representations of finite groups, groups of Lie type, fusion systems, triangulated categories, code theory. This research has a worldwide collaboration with constant supports from NSFC.  


 1. Group and Representation Theory (ZHANG Jiping, WANG Jie, WANG Lizhong)  


The highlights of contributions in Group and representation theory include the existence theorem on defect zero blocks By Jiping Zhang [Zhang, Acta Math. Sin. (1987)] and the classification of linear groups of small degree over fields of finite characteristic by Zhang with Blau [Blau-Zhang, J. Algebra (1993)] and the triangulated equivalences between local and global blocks (categorifications of modular representation theory of finite groups) by Zhang and Lizhong Wang [Wang-Zhang J. Pure Appl. Algebra (2016) and Wang-Zhang, J. Pure Appl. Algebra (to appear)]. They also introduce and investigate the rank of fusion systems [Wang-Zhang, Ann. Mat. Pura Appl. (2016)]. Zhang and Blau’s works also solve Brauer’s problem 39 and 40.
Recent years, Jie Wang investigates the impact of the point stabilizer of primitive permutation groups to the structure of the whole group. For example, he obtained the reduction theorem on symmetric and alternating groups with a doubly transitive subconstituent. Complete classification of primitive groups with certain specified subconstituent are achieved. For example, all primitive groups with solvable 2-transitive subconstituent are determined [Wang, J. Algebra (2013)]. In algebraic graph theory, Jie Wang focus on the graphs with certain symmetric properties, such as Cayley graphs and s-arc-transitive graphs. For example, all 5-arc transitive cubic Cayley graphs on finite simple groups are determined. For symmetric Cayley graphs of valency 6 on finite non-abelian simple groups, we proved that most of such graphs are normal, which guarantees the full automorphism group of the graph has an explicit description [Fang-Wang, JCT (2011)].
The existence and uniqueness proofs of Tits simple group and Dickson group were given by Lizhong Wang in joint work with Michler [Michler-Wang, AC (2008) and Michler-Wang, AC (2011)]. This work is part of the project classifying finite simple groups by using Brauer’s original principle.  

2. Algebraic Number Theory (GAO Xia, WANG Fuzheng)  


In [Gao, JNT (2011)], Xia Gao generalized Datskovsky and Wright’s beautiful formula on the zeta function of integral binary cubic forms to the case of decomposable forms of degree d in d−1 variables associated with a fixed degree d number field. As a by-product, he obtained a characterization of the fractional ideals of a number field K which can be generated by elements from a hyperplane of K. This in turn characterizes the conductor ideals of orders of number fields. He also studied various types of orders of a number field. In particular, he obtained a nice recipe for constructing cubic rings.
In [Gao, JNT (to appear)], he gave a new proof of the Ohno-Nakagawa Theorem by exploiting the rich algebraic structures lying beneath Shintani’s four famous zeta functions. He believed that the techniques developed in this paper, especially the L-series interpretation of Shintani zeta functions, could be used to solve conjectures of this type in other prehomogeneous vector spaces (for example the F4 case) and beyond.  

  3. Lie Algebra (ZHAO Yufeng)  


Major research area lies in the classification theory of representations of simple Lie algebras and Lie superalgebras. These algebraic structures arises in some important physics models, such as string theory and standard model.

With projective differential operators realizations, Yufeng Zhao gave the irreducibility criteria of induced modules of special and orthogonal Lie algebras (joint with Xiaoping Xu). As an application, they gave the calculations of b-funtions. Another part of their research is the classification of Verma modules of exceptional simple linearly compact Lie superalgebra (joint with Victor Kac and Catarrini), the E(5,10) case is done up to now.  


  4. Hessian Polyhedra, Invariant Theory and Appell Hypergeometric PDE (YANG Lei)  


It is well known that Klein's celebrated book on the icosahedron is one of the most important and influential books of 19th-century mathematics. This book has had a long-lasting influence, and is still highly regarded and much used. Klein showed that four apparently disjoint theories: the symmetries of the icosahedron (geometry), the resolution of equations of the fifth degree (algebra), the differential equation of hypergeometric functions (analysis) and the modular equations of elliptic modular functions (arithmetic) are in fact dominated by the structure of a single object, the simple group A_5 of 60 elements. It provides a wonderful example on the fundamental unity of mathematics.  
The book by Lei Yang gives the complex counterpart of Klein's classic book on the icosahedron. It shows that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group G_216. It provides another beautiful example on the fundamental unity of mathematics.  
References: Monograph by Lei Yang, Hessian Polyhedra, Invariant Theory and Appell Hyper- geometric Functions, World Scientific (Singapore), 400 pp., Feb 2018, ISBN: 978-981-3209-47-3.