The Stable Limit DAHA and the Double Dyck Path Algebra

TL;DR

This paper investigates the stable limit of the double affine Hecke algebra (DAHA) of type GL and its connection to the double Dyck path algebra, revealing new algebraic structures and representations in infinite variables.

Contribution

It introduces the +stable limit DAHA and demonstrates its representation on almost symmetric polynomials, linking it to the double Dyck path algebra.

Findings

Established the compatibility of Cherednik operators with inverse systems of polynomial rings.

Constructed the +stable limit DAHA acting on an infinite-variable polynomial space.

Connected the double Dyck path algebra to the standard representation of the stable limit DAHA.

Abstract

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.