Alternating sign matrix: Difference between revisions

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{{distinguish|Alternant matrix}}

[[File:Alternating{{Image Sign Matrices of Size 3.svg|400pxframe|thumbwidth=340|align=right|caption=The seven alternating sign matrices of size 3]]

|content=<math>\begin{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 generalize [[Permutation matrix|permutation matrices]] and 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 P. Robbins|David Robbins]], and Howard Rumsey in the former context.

\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}</math>}}

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 matrix|permutation matrices]] and 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 P. Robbins|David Robbins]], and Howard Rumsey in the former context.<ref>{{citation

| last = Hone | first = Andrew N. W.

| doi = 10.1098/rsta.2006.1887

| issue = 1849

| journal = Philosophical Transactions of the Royal Society of London

| mr = 2317901

| pages = 3183–3198

| title = Dodgson condensation, alternating signs and square ice

| volume = 364

| year = 2006| bibcode = 2006RSPTA.364.3183H

}}</ref> 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 P. Robbins|David Robbins]], and Howard Rumsey in the former context.

==ExampleExamples==

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

A [[permutation matrix]] is an alternating sign matrix, and an alternating sign matrix is a permutation matrix if and only if no entry equals {{math|−1}}.

[[File:Matrice signes alternants 4x4.svg|thumbnail|Puzzle picture]]

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

[[File:Matrice signes alternants 4x4.svg|thumbnail|Puzzle picture]]

:<math>

\begin{bmatrix}

Line 17 ⟶ 71:

</math>

==Alternating sign matrix conjecturetheorem==

The ''alternating sign matrix conjecturetheorem'' states that the number of <math>n\times n</math> alternating sign matrices is

:<math>

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

</math>

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

:[[1 (number)|1]], 1, [[2 (number)|2]], [[7 (number)|7]], [[42 (number)|42]], 429, 7436, 218348, … {{OEIS|id=A005130}}.

This conjecturetheorem was first proved by [[Doron Zeilberger]] in 1992.<ref>Zeilberger, Doron, [http://www.combinatorics.org/Volume_3/Abstracts/v3i2r13.html "Proof of the alternating sign matrix conjecture"], ''[http://www.combinatorics.org/ Electronic Journal of Combinatorics]'' 3 (1996), R13.</ref> In 1995, [[Greg Kuperberg]] gave a short proof<ref>[[Greg Kuperberg|Kuperberg, Greg]], [http://frontarxiv.math.ucdavis.eduorg/abs/math.CO/9712207 "Another proof of the alternating sign matrix conjecture"], ''International Mathematics Research Notes'' (1996), 139-150.</ref> based on the [[Yang-BaxterYang–Baxter equation]] for the six -vertex model with ___domain -wall boundary conditions, that uses a determinant calculation, due to Anatoli Izergin.<ref>"Determinant formula for the six-vertex model", A. G. Izergin et al. 1992 ''J. Phys. A'': Math. Gen. 25 4315.</ref> whichIn solves2005, recurrencea relationsthird dueproof towas given by [[VladimirIlse KorepinFischer]] using what is called the ''operator method''.<ref>V.{{Cite E.journal|last=Fischer|first=Ilse|title=A Korepin,new [http://projecteuclid.org/euclid.cmp/1103921777 Calculationproof of normsthe ofrefined Bethealternating wavesign functions],matrix Comm.theorem|journal=Journal Math.of Phys.Combinatorial Volume 86Theory, Number 3Series (1982), 391-418A|year=2005|volume=114|issue=2|pages=253–264|doi=10.1016/j.jcta.2006.04.004|arxiv=math/0507270|bibcode=2005math......7270F}}</ref>

==Razumov–Stroganov conjectureproblem==

In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model (FPL) and ASMs.<ref>Razumov, A.V., Stroganov Yu.G., [httphttps://arxiv.org/abs/cond-mat/0012141 Spin chains and combinatorics], ''Journal of Physics A'', '''34''' (2001), 3185-3190.</ref>

This conjecture was proved in 2010 by Cantini and Sportiello.<ref> L. Cantini and A. Sportiello, [httphttps://arxiv.org/abs/1003.3376 Proof of the Razumov-Stroganov conjecture]''Journal of Combinatorial Theory, Series A'', '''118 (5)''', (2011) 1549–1574,</ref>

==References==

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==Further reading==

* [[David Bressoud|Bressoud, David M.]], ''Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture'', MAA Spectrum, Mathematical Associations of America, Washington, D.C., 1999.{{ISBN|978-0521666466}}

* [[David Bressoud|Bressoud, David M.]] and Propp, James, [httphttps://www.ams.org/notices/199906/fea-bressoud.pdf How the alternating sign matrix conjecture was solved], ''Notices of the American Mathematical Society'', 46 (1999), 637-646637–646.

* Mills, William H., [[David P. Robbins|Robbins, David P.]], and Rumsey, Howard, Jr., Proof of the Macdonald conjecture, ''Inventiones Mathematicae'', 66 (1982), 73-8773–87.

* Mills, William H., [[David P. Robbins|Robbins, David P.]], and Rumsey, Howard, Jr., Alternating sign matrices and descending plane partitions, ''Journal of Combinatorial Theory, Series A'', 34 (1983), 340-359340–359.

* Propp, James, [httphttps://arxiv.org/abs/math/0208125v1 The many faces of alternating-sign matrices], ''Discrete Mathematics and Theoretical Computer Science'', Special issue on ''Discrete Models: Combinatorics, Computation, and Geometry'' (July 2001).

* Razumov, A. V., Stroganov Yu. G., [httphttps://arxiv.org/abs/math/0104216 Combinatorial nature of ground state vector of O(1) loop model], ''Theor. Math. Phys.'', '''138''' (2004), 333-337333–337.

* Razumov, A. V., Stroganov Yu. G., [http://arxiv.org/abs/cond-mat/0108103 O(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices], ''Theor. Math. Phys.'', '''142''' (2005), 237237–243, {{arxiv|cond-243.mat/0108103}}

* [[David P. Robbins|Robbins, David P.]], The story of <math>1, 2, 7, 42, 429, 7436, \cdotsdots</math>, ''The Mathematical Intelligencer'', 13 (2), 12-1912–19 (1991), {{doi|10.1007/BF03024081}}.

* [[Doron Zeilberger|Zeilberger, Doron]], [http://nyjm.albany.edu:8000/j/1996/2-4.pdf Proof of the refined alternating sign matrix conjecture], ''New York Journal of Mathematics'' 2 (1996), 59-6859–68.

==External links==

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* [http://www.findstat.org/AlternatingSignMatrices Alternating sign matrices] entry in the [http://www.findstat.org/ FindStat] database

{{Matrix classes}}

[[Category:Matrices]]

[[Category:Matrices (mathematics)]]

[[Category:Enumerative combinatorics]]