#### Home>Research>Research Areas>Algebra, Combination and Number Theory

Weekly seminar links: Algebra, Combination and Number Theory

##### 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.