Hochschild cohomology rings of tame Hecke algebras

Let A be a tame Hecke algebra of type A. Based on the minimal projective bimodule resolution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathbb{P} , \delta)}$$\end{document} of A constructed by Schroll and Snashall, we first give an explicit description of the so-called “comultiplicative structure” of the generators of each term Pn in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathbb{P} , \delta)}$$\end{document} , and then apply it to define a chain map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta: \mathbb{P} \rightarrow \mathbb{P} \otimes_A \mathbb{P}}$$\end{document} and thus show that the cup product in the level of cochains for the tame Hecke algebra A is essentially juxtaposition of parallel paths up to sign. As a consequence, we determine the structure of the Hochschild cohomology ring of A under the cup product by giving an explicit presentation by generators and relations.