Permutation representations on invertible matrices

Small maximal spaces of non-invertible matrices

A vector space A of matrices is called rank-critical if any vector space that properly contains A has a strictly higher generic rank. I present a sufficient condition for A to be rank-critical, and apply this condition to prove that certain Lie algebra representations have rank-critical images. This contruction yields, for infinitely many n, an 8-dimensional space of n-times-n matrices that is maximal singular (= rank-critical of rank n-1), while--to the best of my knowledge--the minimal dimension of such a space was hitherto believed to grow with n. As another application, I prove that for an…

LP Decodable Permutation Codes based on Linearly Constrained Permutation Matrices

A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes. Making use of linear constraints imposed on the permutation matrices, we can formulate a minimum Euclidian distance decoding problem for the proposed class of permutation codes as a linear programming (LP) problem. The main feature of this class of permutation codes, called LP decodable permutation codes, is this LP decodability. It is demonstrated that the LP decoding performance of the proposed class of permutation codes is characterized by the vertices of the code polytope of the c…

伴随矩阵、置换与格理想

Companion matrices, permutations and lattice ideals

本文揭示了仿射n维空间中对称二项式理想族的伴随矩阵约化与置换矩阵之间的新联系，并刻画了这些变换所生成的群及其与格理想的关系。

Random Matrices and Random Permutations

We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions of $n$, the 1st, 2nd, and so on, rows behave, suitably scaled, like the 1st, 2nd, and so on, eigenvalues of a Gaussian random Hermitian matrix as $n$ goes to infinity. Our proof is based on an interplay between maps on surfaces and ramified coverings of the sphere. We also establish a connection of this problem with intersection theory on the moduli spaces of curves.

广义置换矩阵与$\frak{sl}(m|1)$上的非权模

Generalized permutation matrices and non-weight modules over $\mathfrak{sl}(m|1)$

本文研究$\frak{sl}(m|1)$上秩$(k|k)$的非权模，构造显式模族，给出同构定理与不可分解性判据。

平面网络与可逆矩阵的多重 Horn 问题

Multiple Horn problems for planar networks and invertible matrices

本文研究平面网络多重 Horn 问题，提出迹等式、菱形不等式和四面体等式作为必要条件，并猜想其在网络足够大时也充分；同时证明乘法问题在大缩放参数下 Duistermaat-Heckman 测度集中于八面体递推轨迹附近。