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談話会・数理科学講演会
過去の記録 ～05/01｜次回の予定｜今後の予定 05/02～
| 担当者 | 会田茂樹，大島芳樹，志甫淳（委員長），高田了 |
|---|---|
| セミナーURL | https://www.ms.u-tokyo.ac.jp/seminar/colloquium/index.html |
2017年07月07日(金)
15:30-16:30 数理科学研究科棟(駒場) 002号室
Richard Stanley 氏 (MIT/University of Miami)
Smith Normal Form and Combinatorics (English)
http://www-math.mit.edu/~rstan/
Richard Stanley 氏 (MIT/University of Miami)
Smith Normal Form and Combinatorics (English)
[ 講演概要 ]
Let $R$ be a commutative ring (with identity) and $A$ an $n \times n$ matrix over $R$. Suppose there exist $n \times n$ matrices $P,Q$ invertible over $R$ for which PAQ is a diagonal matrix $diag(e_1,...,e_r,0,...,0)$, where $e_i$ divides $e_{i+1}$ in $R$. We then call $PAQ$ a Smith normal form (SNF) of $A$. If $R$ is a PID then an SNF always exists and is unique up to multiplication by units. Moreover if $A$ is invertible then $\det A=ua_1\cdots a_n$, where $u$ is a unit, so SNF gives a
canonical factorization of $\det A$.
We will survey some connections between SNF and combinatorics. Topics will include (1) the general theory of SNF, (2) a close connection between SNF and chip firing in graphs, (3) the SNF of a random matrix of integers (joint work with Yinghui Wang), (4) SNF of special classes of matrices, including some arising in the theory of symmetric functions, hyperplane arrangements, and lattice paths.
[ 参考URL ]Let $R$ be a commutative ring (with identity) and $A$ an $n \times n$ matrix over $R$. Suppose there exist $n \times n$ matrices $P,Q$ invertible over $R$ for which PAQ is a diagonal matrix $diag(e_1,...,e_r,0,...,0)$, where $e_i$ divides $e_{i+1}$ in $R$. We then call $PAQ$ a Smith normal form (SNF) of $A$. If $R$ is a PID then an SNF always exists and is unique up to multiplication by units. Moreover if $A$ is invertible then $\det A=ua_1\cdots a_n$, where $u$ is a unit, so SNF gives a
canonical factorization of $\det A$.
We will survey some connections between SNF and combinatorics. Topics will include (1) the general theory of SNF, (2) a close connection between SNF and chip firing in graphs, (3) the SNF of a random matrix of integers (joint work with Yinghui Wang), (4) SNF of special classes of matrices, including some arising in the theory of symmetric functions, hyperplane arrangements, and lattice paths.
http://www-math.mit.edu/~rstan/
