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2-adic valuations of certain ratios of products of factorials and applications

Shmuel Friedland·Christian Krattenthaler
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Abstract

We prove the conjecture of Falikman--Friedland--Loewy on the parity of the degrees of projective varieties of $n\times n$ complex symmetric matrices of rank at most $k$. We also characterize the parity of the degrees of projective varieties of $n\times n$ complex skew symmetric matrices of rank at most $2p$. We give recursive relations which determine the parity of the degrees of projective varieties of $m\times n$ complex matrices of rank at most $k$. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of $n\times n$ skew symmetric real matrices and of $m\times n$ real matrices containing a nonzero matrix of rank at most $k$. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.

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Shmuel Friedland, Christian Krattenthaler. 2005-08-26. 2-adic valuations of certain ratios of products of factorials and applications. https://arxiv.org/abs/math/0508498

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