219:1 On Shabbos during the week of shiva, the mourner observes the private aspects of mourning, i.e., he may not engage in marital relations or wash. Public forms of mourning, however, are not observed. Therefore, before reciting Mizmor Shir l’Yom HaShabbos on Friday evening, the mourner puts on regular shoes, sits on a normal chair, and changes out of his torn clothes. Studying Torah is prohibited, as this is done in private, but the mourner is permitted to review the weekly Torah portion (“shnayim mikra v’echad targum”) because this is an obligation the same as reciting Shema and other parts of the daily service.
219:2 If the mourner was called to the Torah, he must go up. This is because, if he were to refuse, he would be demonstrating public mourning. Rabbeinu Tam was called up every Shabbos for the third aliyah. When he was sitting shiva, the gabbai didn’t call him but he went up anyway. He said, “Since I am called for the third aliyah every week, people seeing that I did not go up today will know that it’s because I’m sitting shiva, which would be considered public mourning.”
Similarly, if a kohein is in mourning but there’s no other kohein in shul, he must be called to the Torah. It’s preferable, however, that he leave the shul before the Torah is removed from the ark. Similarly, if the mourner has a son to be circumcised, the practice is for him to be called to the Torah. He is called as per usual because, were he not to be called, it would be considered mourning in public. It’s preferable, however, that he not be in shul when the Torah is read. Regarding a woman in mourning who goes to shul after giving birth, see 211:11.