Nonvanishing of Kronecker coefficients for rectangular shapes

Details Nonvanishing of Kronecker coefficients for rectangular shapes Nonvanishing of Kronecker coefficients for rectangular shapes

2009-10-23

arxiv.org/abs/0910.4512v3

Advances in Mathematics 227, 2082--2091 (2011)

Mathematics

Group Theory

Added Section 1.1 containing motivation coming from geometric complexity theory. Added a result on symmetric Kronecker coeffic

Scientific paper

10.1016/j.aim.2011.04.012

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.

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