Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)

Extended abstracts listed by author > Guillot Dominique

Schur polynomials and matrix positivity preservers
Alexander Belton  1  , Dominique Guillot  2  , Apoorva Khare  3  , Mihai Putinar  4, 5  
1 : Lancaster University  -  Website
Lancaster University, Bailrigg, Lancaster, LA1 4YW -  Royaume-Uni
2 : University of Delaware [Newark]  -  Website
Newark, DE 19716 USA -  États-Unis
3 : Stanford University [Stanford]  -  Website
450 Serra Mall, Stanford, CA 94305-2004 -  États-Unis
4 : University of California [Santa Barbara]  (UCSB)  -  Website
Santa Barbara, CA 93106 -  États-Unis
5 : Newcastle University [Newcastle]  -  Website
Newcastle upon Tyne NE1 7RU -  Royaume-Uni

A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers. 

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