[Paper Review] Schur partition theorems via perfect crystal

Shunsuke Tsuchioka, Masaki Watanabe|arXiv (Cornell University)|Sep 7, 2016

Algebraic structures and combinatorial models30 references3 citations

TL;DR

This paper introduces two new classes of integer partitions—$ S_p $ and $ \mathsf{Schur}_p $, generalizing Schur's 1926 partition theorem for odd integers $ p \geq 3 $. Using the representation theory of affine Lie algebras and $ A^{(2)}_{p-1} $-crystal structures, it proves that $ S_p $ is partition-theoretically equivalent to sets of odd and strict $ p $-class regular partitions, providing a computer-free proof for Andrews' 3-parameter generalization of the Rogers–Ramanujan identities at $ p=5 $, and reproving Schur's original theorem at $ p=3 $.

ABSTRACT

Motivated by spin modular representations of the symmetric groups, we propose two generalizations of the Schur regular partitions for an odd integer $p\geq 3$. 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 $A^{(2)}_{p-1}$-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.

Motivation & Objective

• Generalize Schur's 1926 partition theorem for odd $ p \geq 3 $ using crystal lattice structures.
• Define two new partition sets: $ S_p $ (within $ p $-strict partitions) and $ \mathsf{Schur}_p $ (within strict partitions), both satisfying a $ p $-Schur inequality and avoiding forbidden subpatterns.
• Establish that $ S_p $ is partition-theoretically equivalent to the sets of odd and strict $ p $-class regular partitions.
• Provide a new, computer-free proof of Andrews' 3-parameter generalization of the Rogers–Ramanujan identities at $ p=5 $, previously proved by Andrews–Bessenrodt–Olsson.
• Connect the partition-theoretic results to spin modular representations of symmetric groups and quantum group theory via affine Lie algebra crystals.

Proposed method

• Define $ p $-Schur inequality: $ \lambda_i - \lambda_{i+h} \geq p $, with strict inequality if $ \lambda_i \in p\mathbb{Z} $, where $ p = 2h+1 $.
• Introduce the set $ \mathsf{Forb}_p $ of forbidden subpatterns (with wildcards $ \ast $) to define $ S_p $ as $ p $-strict partitions avoiding these patterns.
• Construct a crystal structure on $ S_p $ using the $ A^{(2)}_{p-1} $-crystal framework, showing it forms a perfect crystal of level 1.
• Define a bijection $ \Psi^{\mathsf{str}}_1 $ and $ \Psi^{\mathsf{str}}_2 $ to decompose partitions into components in $ \mathsf{Str}^{[p+1,p+h]} $, $ \mathsf{Str}^{[p-h,p-1]} $, etc., using residue-based operations.
• Use the crystal operator $ \tilde{f}_i $ to define raising operators that add boxes of residue $ i \mod p $, ensuring compatibility with the $ p $-Schur inequality.
• Prove that the map $ \beta_p $, which maps components to $ S_p $, is a good crystal morphism, preserving the crystal structure and enabling recursive construction.

Experimental results

Research questions

• RQ1 Can Schur's 1926 partition theorem be generalized to odd integers $ p \geq 3 $ using representation-theoretic tools?
• RQ2 How can the $ A^{(2)}_{p-1} $-crystal structure be used to define and characterize new families of partitions satisfying a generalized $ p $-Schur inequality?
• RQ3 Does the set $ S_p $ of $ p $-strict partitions satisfying the $ p $-Schur inequality and avoiding $ \mathsf{Forb}_p $ subpatterns have the same cardinality as the set of odd or strict $ p $-class regular partitions?
• RQ4 Can the 3-parameter generalization of the Rogers–Ramanujan identities by Andrews be proven without computer assistance using this crystal-theoretic framework?
• RQ5 Is there a uniform crystal-theoretic mechanism that unifies Schur’s original theorem and Andrews’ 3-parameter extension via $ A^{(2)}_{p-1} $-crystals?

Key findings

• For $ p=3 $, the set $ S_3 $ is partition-theoretically equivalent to the sets of odd and strict 3-class regular partitions, thus reproving Schur’s 1926 partition theorem.
• For $ p=5 $, the construction provides a computer-free proof of Andrews’ 3-parameter generalization of the Rogers–Ramanujan identities, confirming a conjecture previously proved by Andrews–Bessenrodt–Olsson.
• The set $ S_p $ forms a perfect $ A^{(2)}_{p-1} $-crystal of level 1, with well-defined crystal operators $ \tilde{f}_i $ that act by adding a box of residue $ i \mod p $, when defined.
• Every partition in $ \mathsf{Schur}_p $ satisfies the $ p $-Schur inequality, and the set $ \mathsf{Schur}_p $ is closed under the crystal operators $ \tilde{f}_i $, ensuring its crystal structure is consistent.
• The map $ \beta_p $ is a good crystal morphism, meaning it preserves the crystal structure and allows the decomposition of $ S_p $ into components that are themselves $ A^{(2)}_{p-1} $-crystals.
• The bijection $ \Psi^{\mathsf{str}}_1 $ and $ \Psi^{\mathsf{str}}_2 $ provide a recursive decomposition of partitions in $ S_p $, enabling the proof of equivalence via component-wise analysis and pattern avoidance.