On subgroup adapted bases for representations of the symmetric group

Abstract

The split basis of an irreducible representation of the symmetric group, $S_{n+m}$, is the basis which is adapted to direct product subgroups of the form $S_n\times S_m$. In this article we have calculated symmetric group subduction coefficients relating the standard Young-Yamanouchi basis for the symmetric group to the split basis by means of a novel version of the Schur-Weyl duality. We have also directly obtained matrix representations in the split basis using these techniques. The advantages of this approach are that we obtain analytical expressions for the subduction coefficients and matrix representations and directly resolve issues of multiplicity in the subduction of $S_n\times S_m$ irreducible representations from $S_{n+m}$ irreducible representations. Our method is applicable to $S_{n+m}$ irreducible representations labelled by Young diagrams with large row length differences. We provide evidence for a systematic expansion in the inverse row length difference and tools to compute the leading contribution, which is exact when row length differences are infinite.

25 pages

References in corpus (2)

• Review of AdS/CFT Integrability: An Overview
• A reduced subduction graph and higher multiplicity in S_n transformation coefficients

Cited by in corpus (4)

• Three-point Functions in $\mathcal{N}=4$ SYM at Finite $N_c$ and Background Independence
• From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications
• Large N anomalous dimensions for large operators in Leigh-Strassler deformed SYM
• Iterative construction of ${\mathfrak{S}}_p\times {\mathfrak{S}}_p$ group-adapted irreducible matrix units for the walled Brauer algebra