Alternating sign matrix

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

\begin{bmatrix} 0&0&1&0\\ 1&0&0&0\\ 0&1&-1&1\\ 0&0&1&0 \end{bmatrix}.

The alternating sign matrix conjecture states that the number of n\times n alternating sign matrices is

\prod_{k=0}^{n-1}\frac{(3k%2B1)!}{(n%2Bk)!} = \frac{1! 4! 7! \cdots (3n-2)!}{n! (n%2B1)! \cdots (2n-1)!}.

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.

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.

References and further reading

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