Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices

• Published in: Journal of combinatorial theory. Series A Vol. 137; pp. 126 - 165
• Main Authors: Betea, D., Wheeler, M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2016
• Subjects: Alternating sign matrices; Cauchy and Littlewood identities; Plane partitions; Six-vertex model; Symmetric functions; Six-vertex model; Cauchy and Littlewood identities; Symmetric functions; Alternating sign matrices; Plane partitions
• ISSN: 0097-3165; 1096-0899
• DOI: 10.1016/j.jcta.2015.08.007

We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2. Alternating sign matrices, subject to certain symmetry properties. The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity, allowing them to be interpreted as generating series for plane partitions. The right hand side of each identity is a partition function of the six-vertex model, on a relevant domain. These can be interpreted as generating series for alternating sign matrices, using the well known bijection with six-vertex model configurations.