Colloquium

Seminar information archive ~03/28Next seminarFuture seminars 03/29~

Organizer(s) ABE Noriyuki, IWAKI Kohei, KAWAZUMI Nariya (chair), KOIKE Yuta
URL https://www.ms.u-tokyo.ac.jp/seminar/colloquium/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)
[ 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 ]
http://www-math.mit.edu/~rstan/