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Quotients of the braid group that are extensions of the symmetric group

Matthew B. Day, Trevor Nakamura

TL;DR

This work addresses extensions of the symmetric group by abelian kernels arising from quotients of the braid group, providing a complete eight-class classification of relevant normal subgroups for $n\ge 4$ and introducing Specht subgroups via kernels in the $M^2$ permutation module. It ties these subgroups to Specht modules for $(n-1,1)$ and $(n-2,2)$, and computes low-dimensional cohomology $H^i(\mathfrak{S}_n;M)$ to understand splitting obstructions; notably, the extension splits only for the Specht subgroup $N_{02}$ when $n$ is odd. The paper also develops explicit descriptions of Specht subgroups (via winding numbers and generating sets), analyzes quotients, and derives precise splitting criteria for extensions arising from Specht subgroups, including level-$m$ reductions. Together these results illuminate the interaction between braid group quotients, symmetric-group representation theory, and cohomological invariants governing semidirect products.

Abstract

We consider normal subgroups $N$ of the braid group $B_n$ such that the quotient $B_n/N$ is an extension of the symmetric group by an abelian group. We show that, if $n\geq 4$, then there are exactly 8 commensurability classes of such subgroups. We define a Specht subgroup to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions $(n-1,1)$ and $(n-2,2)$, working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.

Quotients of the braid group that are extensions of the symmetric group

TL;DR

This work addresses extensions of the symmetric group by abelian kernels arising from quotients of the braid group, providing a complete eight-class classification of relevant normal subgroups for and introducing Specht subgroups via kernels in the permutation module. It ties these subgroups to Specht modules for and , and computes low-dimensional cohomology to understand splitting obstructions; notably, the extension splits only for the Specht subgroup when is odd. The paper also develops explicit descriptions of Specht subgroups (via winding numbers and generating sets), analyzes quotients, and derives precise splitting criteria for extensions arising from Specht subgroups, including level- reductions. Together these results illuminate the interaction between braid group quotients, symmetric-group representation theory, and cohomological invariants governing semidirect products.

Abstract

We consider normal subgroups of the braid group such that the quotient is an extension of the symmetric group by an abelian group. We show that, if , then there are exactly 8 commensurability classes of such subgroups. We define a Specht subgroup to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions and , working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.
Paper Structure (21 sections, 57 theorems, 127 equations)

This paper contains 21 sections, 57 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Acknowledgments
  5. Preliminaries for the classification
  6. Braid groups and symmetric groups
  7. Some permutation and Specht modules
  8. Structure of $M^2_R$
  9. Specht subgroups of $B_n$
  10. Definition and first properties
  11. Proof of the classification
  12. Descriptions of Specht subgroups
  13. Quotients by Specht subgroups
  14. The case when $n=3$
  15. Preliminaries for cohomology
  16. ...and 6 more sections

Key Result

Theorem 1.1

The short exact sequence does not split.

Theorems & Definitions (115)

  • Theorem 1.1: Kordek--Margalit KordekMargalit, Proposition 7.6
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 105 more