Greta Panova

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We prove an equation conjectured by Okada regarding hook-lengths of partitions, namely that 1 n! ∑ λ⊢n f λ ∑ u∈λ r ∏ i=1 (hu − i) = 1 2(r + 1)2 ( 2r r )( 2r + 2 r + 1 ) r ∏ j=0 (n− j), where fλ is the number of standard Young tableaux of shape λ and hu is the hook length of the square u of λ. We also obtain other similar formulas.
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)
The essential cis- and trans-acting elements required for RNA splicing have been defined, however, the detailed molecular mechanisms underlying intron-exon recognition are still unclear. Here we demonstrate that the ratio between stability of mRNA/DNA and DNA/DNA duplexes near 3'-spice sites is a characteristic feature that can contribute to intron-exon(More)
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(More)
The geometric complexity theory program is an approach to separate algebraic complexity classes, more precisely to show the superpolynomial growth of the determinantal complexity dc(per<sub>m</sub>) of the permanent polynomial. Mulmuley and Sohoni showed that the vanishing behaviour of rectangular Kronecker coefficients could in principle be used to show(More)
The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP ws and VNP. Mulmuley and Sohoni [SIAM J Comput 2001] suggested 8to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and(More)
We study the complexity of computing Kronecker coefficients $${g(\lambda,\mu,\nu)}$$ g ( λ , μ , ν ) . We give explicit bounds in terms of the number of parts $${\ell}$$ ℓ in the partitions, their largest part size N and the smallest second part M of the three partitions. When M =  O(1), i.e., one of the partitions is hook-like, the bounds are linear in log(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 Sw by a Schur polynomial sλ (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)