Dimension identities, almost self-conjugate partitions, and BGG complexes for Hermitian symmetric pairs

An almost self-conjugate (ASC) partition has a Young diagram in which each arm along the diagonal is exactly one box longer than its corresponding leg. Classically, the ASC partitions and their conjugates appear in two of Littlewood's symmetric function identities. These identities can be viewed as Euler characteristics of BGG complexes of the trivial representation, for classical Hermitian symmetric pairs. In this paper, we consider partitions in which the arm-leg difference is an arbitrary constant $m$. Regarding these partitions and their conjugates as highest weights, we prove an identity yielding an infinite family of dimension equalities between $\mathfrak{gl}_n$- and $\mathfrak{gl}_{n+m}$-modules. We then interpret this combinatorial result in the context of blocks in parabolic category $\mathcal{O}$: using Enright-Shelton reduction, we find six infinite families of congruent blocks whose corresponding posets of highest weights consist of the partitions in question. These posets, in turn, lead to generalizations of the Littlewood identities and their corresponding BGG complexes. Our results in this paper shed light on the surprising combinatorics underlying the work of Enright and Willenbring (2004).