New Symmetric Plane Partition Identities from Invariant Theory Work of De Concini and Procesi

Nine (= 2 x 2 x 2 + 1) product identities for certain one-variable generating functions of certain families of plane partitions are presented in a unified fashion. The first two of these identities are originally due to MacMahon, Bender, Knuth, Gordon and Andrews and concern symmetric plane partitions. All nine identities are derived from tableaux descriptions of weights of especially nice representations of Lie groups, eight of them for the ‘right end node’ representations of SO˜(2n+1) and Sp(2n). The two newest identities come from a tableaux description which originally arose in work of De Concini and Procesi on classical invariant theory. All of the identities are of the most interest when viewed, in the context of plane partitions with symmetries contained in three-dimensional boxes