We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald processes introduced by Borodin and Corwin. Through this connection, we obtain explicit formulae for the distribution of singular values of a deterministic matrix multiplied by a truncated Haar orthogonal or symplectic matrix under conditions where the latter factor acts as a rank 1 perturbation. Consequently, we generalize the recent Kieburg-Kuijlaars-Stivigny formula for the joint singular value density of a product of truncated unitary matrices to symplectic and orthogonal symmetry classes. Specializing to products of two symplectic matrices with a rank 1 perturbative factor, we show that the squared singular values form a Pfaffian point process.
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Acknowledgements We thank R. Grassucci, D. Bobe, K. Jordan, L. Yen, A. Raczkowski, W. Rice, E. Eng, M. Kopylov, and B. Carragher for access to the electron microscopy facility and help with data collection at the New York Structural Biology Center; and R. Grassucci and Z. Fu for help with data collection at the EM facility at Columbia University. We thank S. Qian and X. Qiang (Pharmaron) for enzyme-activity assay and kinetics studies, M. Geitman (Beactica) for SPR studies, D. Lakshminarasimhan (Xtal BioStructures) for structural studies on the N-terminal segment, proteolysis and early DSF work, Pharmaron for small-molecule synthesis, and H. Blanchette and K. Kreutter (Nimbus) for discussions. The portion of this research that was conducted in the L.T. laboratory was supported by a grant from Nimbus. Some of this work was performed at the Simons Electron Microscopy Center and National Resource for Automated Molecular Microscopy located at the New York Structural Biology Center, supported by grants from the Simons Foundation (349247), NYSTAR, and the NIH National Institute of General Medical Sciences (GM103310).
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