BC_n-对称阿贝尔函数

Abelian sections of the symmetric groups with respect to their index

We show the existence of an absolute constant $α>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and for every $H \leqslant G$ of index at least $3$, one has $|H/[H,H]| \leq |G:H|^{α/ \log \log |G:H|}$. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.

On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.

对称函数与阿贝尔群的显式矩问题

Symmetric functions and the explicit moment problem for abelian groups

本文利用Macdonald多项式性质，给出了随机阿贝尔群分布公式的另一种证明方法。

对称群 $S_n, n\geq 4$ 以及有限非阿贝尔单群不能嵌入任何里奥丹群

The symmetric groups $S_n, n\geq 4$, and finite non-abelian simple groups are not embeddable in any Riordan group

该研究证明了度数大于三的对称群及有限非阿贝尔单群无法嵌入任何里奥丹群，并展示了截断里奥丹群的可解性与无限里奥丹群的不可解性对比，同时给出了交替群 $A_4$ 的显式嵌入结果。

涉及有限阿贝尔群与对称群圈积的 Gelfand 对

Gelfand pairs involving the wreath product of finite abelian groups with symmetric groups

本文推广了对称群的Gelfand对理论，证明了对于有限群G，其与对称群的圈积构成Gelfand对的充分必要条件是G为阿贝尔群。

Nilpotent dessins: Decomposition theorem and classification of the abelian dessins

A map is a 2-cell decomposition of an orientable closed surface. A dessin is a bipartite map with a fixed colouring of vertices. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges, and a regular dessin is symmetric if it admits an additional external symmetry transposing the vertex colours. Regular dessins with nilpotent automorphism groups are investigated. We show that each such dessin is a parallel product of regular dessins whose automorphism groups are the Sylow subgroups. Regular and symmetric dessins with abelian automorph…

对称序列及其他大群的组合性质

Symmetric sequencings and other combinatorial properties of large groups

本文证明了关于对称序列和2序列的两个猜想在大群中成立，并探讨了双序列在阿贝尔群和非阿贝尔群中的扩展结果。

关于熵的 Marton 型不等式以及与 Abel 群陪集的小对称差

On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups

本文利用一个熵的 Marton 型不等式作为黑箱，证明 Abel 群中有限子集的和分布若接近均匀，则其与某有限陪集仅有小对称差，且界在对数因子内最优。

具有消失阿贝尔化的齐次对称迹正六次多项式是埃尔米特平方与交换子的和

Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators

本文在阿贝尔化消失的双变量对称齐次六次多项式空间中，证明迹正性等价于属于埃尔米特平方和的锥，给出了该空间中两类锥的完整显式描述。

Counting Free Abelian Actions

We consider the problem of counting commuting r-tuples of elements of the symmetric group S_n, i.e. computing |Hom(Z^r,S_n)|. The cases r=1,2 are well-known; a product formula for the case r=3 was conjectured by Adams-Watters and later proved by Britnell. In this note we solve the problem for arbitrary r.

On a non-abelian Balog-Szemeredi-type lemma

We show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.

具有对称完美配对的p群上的随机游走

A random walk on p-groups with a symmetric perfect pairing

该研究分析具有对称完美配对的p群上的随机游走，证明随机对称p进矩阵相关过程为马尔可夫链，其生成算子可明确描述且关于Cohen-Lenstra型测度可逆。

广义对称函数

Generalized symmetric functions

众所周知，在无限域上，有限个变量的对称函数环同构于矩阵上的多项式函数的环，矩阵通过一般线性群的共轭作用是不变的。我们推广了这个结果，证明了自由结合代数上固定阶对称张量代数的阿贝尔化同构于无限域上多个矩阵的多项式不变量代数或整数代数.在证明主要结果的同时，给出了任意交换环上代数的交换化除幂的生成元和关系。

度数为7的非交换单群上的对称Cayley图

Symmetric Cayley graphs on non-abelian simple groups of valency 7

该研究改进了非正规7度对称Cayley图的已有刻画，证明其自同构群含特定弧传递非交换单正规子群，明确了群对结构与最大可解正规子群相关性质。