Determinants and alternating sign matrices

Abstract

Let M be an n by n matrix. By a connected minor of M of size k we mean a minor formed from k consecutive rows and k consecutive columns. We give formulas for det M in terms of connected minors, one involving minors of two consecutive sizes and one involving minors of three consecutive sizes. The formulas express det M as sums indexed by sets of alternating sign matrices. These matrices are described here and by W. H. Mills, D. P. Robbins, and H. Rumsey, Jr. ( Invent. Math. 66 (1982), 73–87; J. Combin. Theory Ser. A. 34 (1983), 340–359) . The former study has led to the solution of Macdonald's conjecture on cyclically symmetric plane partitions ( G. E. Andrews, Invent. Math. 53 (1979), 193–225 ; I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” p. 53, Oxford Univ. Press (Clarendon), Oxford, 1979).