Three alternating sign matrix identities in search of bijective proofs (Q5956767)

An alternating sign matrix is a square matrix of entries from \(\{-1,0,1\}\) with the property that in any row or column the entries sum to 1 and the non-zero entries alternate in sign. This paper discusses 3 known identities involving alternating sign matrices, usually interpreting the matrices in terms of the six vertex (square ice) model of statistical mechanics. The identities also involve a one parameter generalisation of determinants known as \(\lambda\)-determinants.NEWLINENEWLINENEWLINEThe author expresses his conviction that bijective proofs of the 3 identities could be found. This hope has rapidly been realised in the case of the second identity, which relates alternating sign matrices to the number of upsets in tournaments. Motivated by an early draft of the work being reviewed, Chapman has found a bijective proof of Bressoud's identity. Conveniently, Chapman's proof appears in the same volume; see \textit{R. Chapman} [Adv. Appl. Math. 27, No. 2-3, 318-335 (2001; Zbl 0990.05002)].