Alternating sign matrix
Encyclopedia
In mathematics
, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices arise naturally when using Dodgson condensation
to compute a determinant. They are also closely related to the six vertex model with domain wall boundary conditions from statistical mechanics
. They were first defined by William Mills, David Robbins
, and Howard Rumsey in the former context.
For example, the permutation matrices are alternating sign matrices, as is
The alternating sign matrix conjecture states that the number of
alternating sign matrices is
This conjecture was first proved by Doron Zeilberger
in 1992. In 1995, Greg Kuperberg
gave a short proof based on the Yang-Baxter equation
for the six vertex model with domain wall boundary conditions, that uses a determinant due to Anatoli Izergin, which solves recurrence relations due to Vladimir Korepin
.
This conjecture was proved in 2010 by Cantini and Sportiello.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices arise naturally when using Dodgson condensation
Dodgson condensation
In mathematics, Dodgson condensation is a method of computing the determinants of square matrices. It is named for its inventor Charles Dodgson...
to compute a determinant. They are also closely related to the six vertex model with domain wall boundary conditions from statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. They were first defined by William Mills, David Robbins
David P. Robbins
David P. Robbins was an American mathematician. He is most famous for introducing alternating sign matrices and his work on generalized Heron's formula....
, and Howard Rumsey in the former context.
For example, the permutation matrices are alternating sign matrices, as is
The alternating sign matrix conjecture states that the number of
alternating sign matrices isThis conjecture was first proved by Doron Zeilberger
Doron Zeilberger
Doron Zeilberger is an Israeli mathematician, known for his work in combinatorics.He is a Board of Governors Professor of Mathematics at Rutgers University...
in 1992. In 1995, Greg Kuperberg
Greg Kuperberg
Greg Kuperberg is an American mathematician of Polish birth known for his contributions to geometric topology, quantum algebra, and combinatorics. Kuperberg is a professor of mathematics at the University of California, Davis....
gave a short proof based on the Yang-Baxter equation
Yang-Baxter equation
The Yang–Baxter equation is an equation which was first introduced in the field of statistical mechanics. It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1971...
for the six vertex model with domain wall boundary conditions, that uses a determinant due to Anatoli Izergin, which solves recurrence relations due to Vladimir Korepin
Vladimir Korepin
Vladimir Korepin is a Russian-American physicist and mathematician. He is a professor at the C. N. Yang Institute of Theoretical Physics of the State University of New York at Stony Brook...
.
Razumov–Stroganov conjecture
In 2001 A.Razumov and Y.Stroganov conjectured a connection between O(1) loop model, fully packaged loop model (FPL) and ASMs.This conjecture was proved in 2010 by Cantini and Sportiello.
External links
- Alternating sign matrix entry in MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...