Alternating sign matrix

{\begin{matrix}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\qquad {\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}}\\{\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}}\qquad {\begin{bmatrix}0&1&0\\1&-1&1\\0&1&0\end{bmatrix}}\qquad {\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}\\{\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}\qquad {\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}\end{matrix}}

The seven alternating sign matrices of size 3

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 generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant.^[1] 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.

Examples

A permutation matrix is an alternating sign matrix, and an alternating sign matrix is a permutation matrix if and only if no entry equals $-1$ .

An example of an alternating sign matrix that is not a permutation matrix is

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

Alternating sign matrix theorem

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

\prod _{k=0}^{n-1}{\frac {(3k+1)!}{(n+k)!}}={\frac {1!\,4!\,7!\cdots (3n-2)!}{n!\,(n+1)!\cdots (2n-1)!}}.

The first few terms in this sequence for n = 0, 1, 2, 3, … are

1, 1, 2, 7, 42, 429, 7436, 218348, … (sequence A005130 in the OEIS).

This theorem was first proved by Doron Zeilberger in 1992.^[2] In 1995, Greg Kuperberg gave a short proof^[3] based on the Yang–Baxter equation for the six-vertex model with domain-wall boundary conditions, that uses a determinant calculation due to Anatoli Izergin.^[4] In 2005, a third proof was given by Ilse Fischer using what is called the operator method.^[5]

Razumov–Stroganov problem

In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model (FPL) and ASMs.^[6] This conjecture was proved in 2010 by Cantini and Sportiello.^[7]

References

^ Hone, Andrew N. W. (2006), "Dodgson condensation, alternating signs and square ice", Philosophical Transactions of the Royal Society of London, 364 (1849): 3183–3198, Bibcode:2006RSPTA.364.3183H, doi:10.1098/rsta.2006.1887, MR 2317901
^ Zeilberger, Doron, "Proof of the alternating sign matrix conjecture", Electronic Journal of Combinatorics 3 (1996), R13.
^ Kuperberg, Greg, "Another proof of the alternating sign matrix conjecture", International Mathematics Research Notes (1996), 139-150.
^ "Determinant formula for the six-vertex model", A. G. Izergin et al. 1992 J. Phys. A: Math. Gen. 25 4315.
^ Fischer, Ilse (2005). "A new proof of the refined alternating sign matrix theorem". Journal of Combinatorial Theory, Series A. 114 (2): 253–264. arXiv:math/0507270. Bibcode:2005math......7270F. doi:10.1016/j.jcta.2006.04.004.
^ Razumov, A.V., Stroganov Yu.G., Spin chains and combinatorics, Journal of Physics A, 34 (2001), 3185-3190.
^ L. Cantini and A. Sportiello, Proof of the Razumov-Stroganov conjectureJournal of Combinatorial Theory, Series A, 118 (5), (2011) 1549–1574,

External links

Alternating sign matrix entry in MathWorld
Alternating sign matrices entry in the FindStat database

[1] Hone, Andrew N. W. (2006), "Dodgson condensation, alternating signs and square ice", Philosophical Transactions of the Royal Society of London, 364 (1849): 3183–3198, Bibcode:2006RSPTA.364.3183H, doi:10.1098/rsta.2006.1887, MR 2317901

[2] Zeilberger, Doron, "Proof of the alternating sign matrix conjecture", Electronic Journal of Combinatorics 3 (1996), R13.

[3] Kuperberg, Greg, "Another proof of the alternating sign matrix conjecture", International Mathematics Research Notes (1996), 139-150.

[4] "Determinant formula for the six-vertex model", A. G. Izergin et al. 1992 J. Phys. A: Math. Gen. 25 4315.

[5] Fischer, Ilse (2005). "A new proof of the refined alternating sign matrix theorem". Journal of Combinatorial Theory, Series A. 114 (2): 253–264. arXiv:math/0507270. Bibcode:2005math......7270F. doi:10.1016/j.jcta.2006.04.004.

[6] Razumov, A.V., Stroganov Yu.G., Spin chains and combinatorics, Journal of Physics A, 34 (2001), 3185-3190.

[7] L. Cantini and A. Sportiello, Proof of the Razumov-Stroganov conjectureJournal of Combinatorial Theory, Series A, 118 (5), (2011) 1549–1574,

[1]

[2]

[3]

[4]

[5]

[6]

[7]

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices (mathematics)

Alternating sign matrix

Hub talks

Alternating sign matrix

Alternating sign matrix

Alternating sign matrix

Hub overview

Add your contribution

Inside this hub

Media Collections

Timelines

Articles

Notes collections

Notes

Days in Chronicle

Root Wikipedia Article

Alternating sign matrix

Examples

Alternating sign matrix theorem

Razumov–Stroganov problem

References

Further reading

External links

Add your contribution

Related Hubs