During the Chazzan's repetition of the Shemoneh Esrei, one is generally required to pay absolute attention and avoid learning etc.
What about when the Chazzan is reciting piyutim during his repetition? Need one focus on his words, or can one learn or engage in other activities? After all, piyutim are not required (the repetition's valid if they're omitted), and there's less of a problem of berachah levatalah.
I witnessed the Rosh Yeshiva of Ner Israel several times pick up a sefer and learn when the Chazzan would choose a slow tune for Mimkomcha of the Shabbos Shacharis Kedusha. I don't see piyutim being any different than the sections between the primary lines of Kedusha, and certainly not any more stringent.
The Rosh Yeshiva never learned or did anything other than stand and listen attentively during the rest of the Chazzan's repetition, and in his shmuessen he publicly decried such practice, so it seems he was not making a special "I'm a Rosh Yeshiva" exception, if such a thing exists.
Shulchan Aruch Harav Orach Chaim 68:1 (basically paraphrasing the Ramo, with a reference to the Maharil):
ומכל מקום לא יעסוק בשום דבר ואפילו בדברי תורה אסור להפסיק לעסוק כל זמן שהצבור אומרים פיוטים כ"ש לדבר שום שיחה בטילה ומכל מקום מי שלומד על ידי הרהור שרואה בספר ומהרהר אין בזה איסור כלל שהרהור אינו כדבור ואינו חשוב הפסק אלא שיש לחוש להמון עם שאל יראו מן הלומד בספר ויפסיקו גם כן בדברים אחרים על כן אין לאדם לפרוש עצמו מהצבור במקום שנהגו לאמרם ויאמר אותם עמהם
Nevertheless do not occupy yourself with anything, even words of Torah are forbidden to interrupt and do the whole time that the congregation is saying Piyutim even more so any idle speech nevertheless someone who learns through thinking about what he is seeing in the book there is nothing forbidden at all because thinking is not like speaking and is not an interruption, but you should be concerned about the common people that they should not see the one learning in a book and they will also interrupt also for other things. Therefore no person should separate himself from the congregation in a place where they are accustomed to say them and he should say them with him.
Now the context there is Piyutim that some have a custom to add during Birchas Kriyas Shema, so the concern about an interruption isn't there, but the impression that it leaves on the congregation certainly is. Especially (as is explained further in that siman) where you are talking about Piyutim in Chazaras HaShatz which are not controversial (those in Birchas Krias Shema the Mechabar says to stop the custom of saying them) and common people who won't make a distinction about which words constitute the required part of the repetition and which words don't.
