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Schur partition theorems via perfect crystal

Shunsuke Tsuchioka, Masaki Watanabe

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.

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 . One forms a subset of the set of -strict partitions, and the other forms that of strict partitions. We prove that each set has a basic -crystal structure. For , it reproves Schur's 1926 partition theorem, a mod 6 analog of Rogers-Ramanujan partition theorem (RRPT). For , 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.

Paper Structure

This paper contains 39 sections, 43 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main results
  3. Andrews' 3-parameter generalization of RRPT
  4. Schur PT and $p$-modular representations of the Schur cover $\widehat{\mathfrak{S}}_{n}$
  5. Modular branching rules and Kashiwara crystal theory
  6. A definition of $\mathsf{Schur}_p$ and future directions
  7. Notations and Conventions
  8. Combinatorics of $S_p$
  9. Definition and routine checks
  10. The map $\lambda\mapsto(\lambda^+,\lambda^-)$
  11. Zigzag property
  12. Proof of ${{\mu}\sqcup{\lambda^+}}\not\in S_p$ implies ${{\mu}\sqcup{\lambda}}\not\in S_p$
  13. Proof of ${{\mu}\sqcup{\lambda}}\not\in S_p$ implies ${{\mu}\sqcup{\lambda^+}}\not\in S_p$
  14. Setting
  15. The case $\lambda_b=jp+c$ for $1\leq c\leq h$
  16. ...and 24 more sections

Key Result

Theorem 1.4

For an odd integer $p\geq 3$, we have $S_p\stackrel{\mathsf{PT}}{\sim}D^{\mathsf{odd}}_{p}$ (resp. $S_p\stackrel{\mathsf{PT}}{\sim}D^{\mathsf{str}}_{p}$).

Figures (1)

  • Figure 1: The diagrams $A^{(2)}_{\textrm{even}}$, $(A^{(2)}_{\textrm{even}})^{\dagger}$, $D_\ell$, $D^{(2)}_{n+1}$ ($n\geq 2, \ell\geq 4$)

Theorems & Definitions (101)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: An3
  • Theorem 1.4
  • Theorem 1.5: RRPT
  • Theorem 1.6: An4
  • Theorem 1.7: An4$=$ABO
  • Theorem 1.8: Sch
  • Theorem 1.9: BMO$=$ABO
  • Theorem 1.10: BK
  • ...and 91 more