Bounds on the Kronecker coefficients

Abstract

We present several upper and lower bounds on the Kronecker coefficients of the symmetric group. We prove $k$-stability of the Kronecker coefficients generalizing the (usual) stability, and giving a new upper bound. We prove a lower bound via the characters of $S_n$. We apply these and other results to generalize Sylvester's unimodality of the $q$-binomial coefficients ${\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_q$ as polynomials in $q$: we derive explicit sharp bounds on the differences of their consecutive coefficients.

18 pages; v2 -- updated and added references

References in corpus (4)

• On the complexity of computing Kronecker coefficients
• Kronecker products, characters, partitions, and the tensor square conjectures
• Counterexamples to Okounkov's Log-Concavity Conjecture
• A combinatorial proof of strict unimodality for $q$-binomial coefficients