A formula for the partial trace of a full symmetrizer is obtained. The formula is used to provide an inductive proof of the well-known formula for the dimension of a full symmetry class of tensors.
In this paper, we first present simple proofs of Chois results [4], then we give a short alternative proof for Fiedler and Markhams inequality [6]. We also obtain additional matrix inequalities related to partial determinants.
We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about the genuine $L^2$-automorphic spectrum, into a linear combination of stable trace formulas of products of split odd orthogonal groups via endoscopic transfer. By adapting the strategies of Arthur and Moeglin-Waldspurger from the linear case, the proof is built on a long induction process that mixes up local and global, geometric and spectral data. As a by-product, we also stabilize the local trace formula for metaplectic groups over any local field of characteristic zero.
Through combining the work of Jean-Loup Waldspurger (cite{waldspurger10} and cite{waldspurgertemperedggp}) and Raphael Beuzart-Plessis (cite{beuzart2015local}), we give a proof for the tempered part of the local Gan-Gross-Prasad conjecture (cite{ggporiginal}) for special orthogonal groups over any local fields of characteristic zero, which was already proved by Waldspurger over $p$-adic fields.
In this work the notions of partial action of a weak Hopf algebra on a coalgebra and partial action of a groupoid on a coalgebra will be introduced, just as some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of partial action on a coalgebra and partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.
The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family of linear mappings from matrix-valued inputs to matrix-valued outputs; this model subsumes the trace regression model and thus the linear regression model. Borrowing tools from quantum information theory, where partial trace operators have been extensively studied, we propose a framework for learning partial trace regression models from data by taking advantage of the so-called low-rank Kraus representation of completely positive maps. We show the relevance of our framework with synthetic and real-world experiments conducted for both i) matrix-to-matrix regression and ii) positive semidefinite matrix completion, two tasks which can be formulated as partial trace regression problems.