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We prove strict unimodality of the q-binomial coefficients (n k) q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of Sn representations. Résumé. Nous prouvons l'unimodalité stricte des coefficients q-binomiaux (n k) q comme des polynômes en q. La preuve est basée(More)
We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups Sn contain all irreducibles as their constituents. Our main result is that they contain representations corresponding to hooks and two row Young diagrams. For that, we develop a new sufficient condition for the positivity(More)
We present new proofs and generalizations of unimodality of the q-binomial coefficients n k q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker coefficients of representations of Sn. Other applications of this approach include strict unimodality of the diagonal q-binomial(More)
We study multiplication of any Schubert polynomial S w by a Schur polynomial s λ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions λ, including hooks and the 2 × 2(More)
We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial. Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the(More)
We prove an equation conjectured by Okada regarding hook-lengths of partitions, namely that 1 n! λ⊢n f 2 λ u∈λ r i=1 (h 2 u − i 2) = 1 2(r + 1) 2 2r r 2r + 2 r + 1 r j=0 (n − j), where f λ is the number of standard Young tableaux of shape λ and h u is the hook length of the square u of λ. We also obtain other similar formulas.
The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VPws and VNP. Mulmuley and Sohoni [34] suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent(More)
  • Jang Soo Kim, Karola Mészáros, Greta Panova, David B Wilson
  • 2012
Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We give two bijections between " cover-inclusive " Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to(More)