nLab alternating sign matrix

Definition

A square matrix with integer entries is an alternating sign matrix if all of the following holds

• entries may be $0,1,-1$ only

• the sum of each row and of each column is $1$

• the nonzero entries in each row and each column alternate in sign

Examples

• every permutation matrix

• the following matrix:

  $$\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 &-1 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right)$$

Literature

• Wikipedia: Alternating sign matrix

Introduced in

• W. H. Mills, D. P. Robbins, H. Rumsey. Jr., Proof of the Macdonald conjecture, Invent. Math. 66 (1982) 73–87 doi
• W. H. Mills, D. P. Robbins, H. Rumsey. Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983) 340–359 doi

On Alternating Sign Matrix Conjecture

• David Bressoud, Proofs and confirmations: the story of the Alternating Sign Matrix Conjecture, Cambridge University Press (1999) [doi:10.1017/CBO9780511613449]

Appearance in cancellation of terms in Dodgson condensation

• Andrew N. W. Hone, Dodgson condensation, alternating signs and square ice, Phil. Trans. R. Soc. A (2006) 364, 3183–3198 doi

• D. P. Robbins, H. Rumsey, Determinants and alternating-sign matrices, Advances in Math. 62 (1986) 169–184 doi

• J. Propp, The many faces of alternating-sign matrices, in Discrete Models: Combinatorics, Computation, and Geometry, Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discr´et. (Paris, 2001) 43–58

category: algebra, combinatorics