Schur partition theorems via perfect crystal

Abstract

Motivated by spin modular representations of the symmetric groups, we propose two generalizations of the Schur regular partitions for an odd integer $p\ge 3$. One forms a subset of the set of $p$-strict partitions, and the other forms that of strict partitions. We prove that each set has a basic $A_{p-1}^{(2)}$-crystal structure. For $p=3$, it reproves Schur's 1926 partition theorem, a mod 6 analog of Rogers-Ramanujan partition theorem (RRPT). For $p=5$, it gives a computer-free proof of a conjecture by Andrews during his 3-parameter generalization of RRPT, which was first proved by Andrews-Bessenrodt-Olsson.

25 pages

References in corpus (2)

• A framework of Rogers-Ramanujan identities and their arithmetic properties
• Young Walls of Type $D^{(}2{)}_n+1$ and Strict Partitions