You can certainly try asking, but I expect you won't get a useful answer. In the cases I'm familiar with (mathematics departments in the U.S.), you aren't likely to get any official answer beyond "no comment". This isn't because the department is being unhelpful or deliberately withholding information they could reasonably share. Instead, it's because offering useful feedback is really hard.
Rejected applicants sometimes imagine that the rejection may have been due to some identifiable flaw they could fix, but in my experience this is rarely the case. Most rejections are based on an overall evaluation, rather than isolated flaws. Even if a single problem derailed the application, there might have been other issues that were never identified because discussion stopped when the application died. Nobody wastes times on hypotheticals like "Would the candidate have been admitted if it weren't for this problem?", since the committee is eager to move on to the more contentious cases. So even if the committee can point to a specific problem, they generally can't predict that fixing it would lead to admission. It's not particularly useful or encouraging to be told "Here's a serious problem with your application, but don't assume you'll get in if you fix it. We haven't even considered that question, so who knows?" (Plus there are legal issues: you don't want to be sued by someone who is told their application was rejected for a specific reason and then discovers that you admitted another applicant with the same issue. You might have had an excellent reason - perhaps the rest of the other application was magnificent - but it could still look like a pretext for discrimination. It's safest not to offer any explanation that oversimplifies the full decision making process.)
These cases with isolated flaws are really not typical. Instead, most feedback would fall into two categories: "Your application just wasn't good enough overall for us to even consider you" or "Your application was good and you made the short list, but you were outcompeted by people with even stronger applications". The first sort of feedback adds insult to injury, and it's wise to avoid making enemies among the applicants you reject. The second sort of feedback is also not constructive, since there's usually no good way to convey to the applicant how the final comparison was made. The evaluation process consisted of multiple readers, perhaps with differing opinions, followed by a committee discussion that compared the application with the competition. Confidential letters of recommendation played a major role, and in any case the process was lengthy and involved enough that it's difficult to summarize it usefully.
For comparison, U.S. National Science Foundation review panels provide applicants with copies of the written reviews and ratings of their proposals, together with a brief summary of the panel discussion. This is about as much transparency as one could reasonably hope for, but it rarely leaves rejected applicants with a clear idea of how to improve their future proposals. In many cases they don't even have a good understanding of why they didn't make it (beyond "some other proposals were even more compelling").