We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/– stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation of the +stable limit DAHA.

Over years of development in secure multi-party computation (MPC), many sophisticated functionalities have been made practical, and multi-dimensional operations occur more and more frequently in MPC protocols, especially in protocols involving datasets of vector elements, such as privacy-preserving biometric identification and privacy-preserving machine learning. In this paper, we introduce a new kind of correlation, called tensor triples, which is designed to make multi-dimensional MPC protocols more efficient. We will discuss the generation process, the usage, and the applications of tensor triples and show that they can accelerate privacy-preserving biometric identification protocols, such as FingerCode, Eigenfaces, and FaceNet, by more than 1000 times, with reasonable offline costs, and grant pre-computability for the secure matrix multiplication process in privacy-preserving machine learning protocols, such as SecureML and SecureNN, while achieving similar efficiency.

We prove a number of results about the structure of the standard representation of the stable limit double affine Hecke algebra (DAHA). More precisely, we address the triangularity, spectrum, and eigenfunctions of the limit Cherednik operators, and construct several bases of Poincaré-Birkhoff-Witt (PBW) type for the stable limit DAHA. We establish a remarkable triangularity property concerning the contribution of certain special elements of the PBW basis of a finite rank DAHA of high enough rank to the PBW expansion of a PBW basis element of the stable limit DAHA. The triangularity property implies the faithfulness of the standard representation. This shows that the algebraic structure defined by the limit operators associated to elements of the finite rank DAHAs is precisely the stable limit DAHA.

We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/- stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation of the +stable limit DAHA.