Nonvanishing of Kronecker coefficients for rectangular shapes

Abstract

We prove that for any partition $({\lambda }_1,...,{\lambda }_{d^2})$ of size $\ell d$ there exists $k\ge 1$ such that the tensor square of the irreducible representation of the symmetric group $S_{k\ell d}$ with respect to the rectangular partition $(k\ell ,...,k\ell )$ contains the irreducible representation corresponding to the stretched partition $(k{\lambda }_1,...,k{\lambda }_{d^2})$. We also prove a related approximate version of this statement in which the stretching factor $k$ is effectively bounded in terms of $d$. This investigation is motivated by questions of geometric complexity theory.

Added Section 1.1 containing motivation coming from geometric complexity theory. Added a result on symmetric Kronecker coefficients. Fixed a bug that made the paper appear twice in v3

References in corpus (5)

• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Quantum marginal problem and representations of the symmetric group
• Uncertainty, Monogamy, and Locking of Quantum Correlations
• Even Partitions in Plethysms
• An overview of mathematical issues arising in the Geometric complexity theory approach to VP v.s. VNP

Cited by in corpus (14)

• On vanishing of Kronecker coefficients
• Eigenvalue Distributions of Reduced Density Matrices
• Even Partitions in Plethysms
• An overview of mathematical issues arising in the Geometric complexity theory approach to VP v.s. VNP
• Inequalities for Moment Cones of Finite-Dimensional Representations
• Unifying and generalizing known lower bounds via geometric complexity theory
• Computing Multiplicities of Lie Group Representations
• A Study of the representations supported by the orbit closure of the determinant
• Permanent versus determinant: not via saturations
• Integrality, Duality and Finiteness in Combinatoric Topological Strings
• The quantum detection of projectors in finite-dimensional algebras and holography
• All-orders asymptotics of tensor model observables from symmetries of restricted partitions
• Permanent versus determinant, obstructions, and Kronecker coefficients
• Maximal lexicographic spectra and ranks for states with fixed uniform margins