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Colloquium
Seminar information archive ～04/30｜Next seminar｜Future seminars 05/01～
| Organizer(s) | AIDA Shigeki, OSHIMA Yoshiki, SHIHO Atsushi (chair), TAKADA Ryo |
|---|---|
| URL | https://www.ms.u-tokyo.ac.jp/seminar/colloquium_e/index_e.html |
2017/07/07
15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Richard Stanley (MIT)
Smith Normal Form and Combinatorics (English)
http://www-math.mit.edu/~rstan/
Richard Stanley (MIT)
Smith Normal Form and Combinatorics (English)
[ Abstract ]
Let R be a commutative ring (with identity) and A an n x n matrix over R. Suppose there exist n x 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.
[ Reference URL ]Let R be a commutative ring (with identity) and A an n x n matrix over R. Suppose there exist n x 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/
