When you are a beloved employee’s replacement

August 2, 2012, 3:25 PM UTC

FORTUNE — Dear Annie: I recently started a new job in the small communications department of a big company. I’m happy to be here, but there’s just one problem. The person who had this job before me was extremely well-liked and left behind some hard-core loyalists, including the person I share office space with. I’ll call her Q. So far, Q has gone out of her way to undermine me. For instance, she has played dumb on tasks she would ordinarily be responsible for, leaving me to figure things out on my own.

I really don’t appreciate her passive-aggressive resentment of me, since her friend left the job through no fault of mine. Also, I hate petty office politics. I just want to do my job and get along with everyone. I realize I need to be patient and give people (especially Q) time to accept me, but I don’t want to be a doormat in the meantime. How do I stick up for myself without being confrontational about it? — New Kid on the Block

Dear New Kid: “It would be interesting to hear Q’s side of this,” muses Monica Wofford, head of Orlando-based executive development firm Contagious Companies, which counts Microsoft (MSFT), United HealthCare, AT&T (T), and SeaWorld among its clients. “Often, we walk around assuming that everyone communicates the same way we do. But they don’t. Q may be assuming, for example, that if you want her help figuring things out, you’ll say so. Meanwhile, it sounds as if no communication is happening at all.”

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Wofford wrote a book called Make Difficult People Disappear: How to Deal With Stressful Behavior and Eliminate Conflict. The title, she notes, refers not to literally making anyone vanish (no matter how much you may wish you had a magic wand), but to training yourself to “stop seeing differences as difficulties. Most of the time, once you make the effort to understand more and change your own expectations, ‘difficult’ people become a lot easier to deal with.”

Great, but how do you do that? Wofford suggests starting with these four steps:

1. Withhold assumptions. Assuming anything about someone else’s behavior, especially someone you don’t know well, is risky and can lead to needless conflict, Wofford says. “Is Q really ‘playing dumb’, or could there be some other reason why she appears to be holding back on information?” Wofford wonders. “Until you discuss the situation with her, you’re only working with your own data, and there are probably some crucial pieces missing.” Withholding assumptions about what’s going on “gives you time to gather more information, so that your perceptions are more accurate.”

2. Hold off on labeling. You barely know your new colleague, yet you’ve already decided she’s a “hard-core loyalist” and “passive-aggressive,” Wofford observes. “These labels aren’t helpful. The trouble is, once you’ve labeled someone, the description tends to stick,” she says.

The human brain is wired in such a way that, once we have characterized someone or something in our minds, “everything they do, or don’t do, will reinforce that,” Wofford adds. “Your brain looks for evidence to support the label and overlooks all the input that runs counter to it — even if there is more of the latter kind of data. You’ll find that, if you ditch the labels, you can be a lot more objective and open-minded,” which will be a big help in the next two steps.

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3. Show an interest. Since, as coworkers, you and Q are both working toward a common goal — i.e., getting work done — why not try to find out from her what you can do to make that process run more smoothly? “You also have to be clear about what you need from her in order to do your job,” says Wofford. “But it’s important to listen carefully. Stay engaged in a dialogue and be genuinely curious. You may be surprised at what you discover.”

4. Ask the right questions. “One question that can be very revealing is, ‘Can you tell me your expectations of our working relationship?’” Wofford says. “Do watch your tone, because ‘expectations’ can be a loaded word. But, asked in a non-confrontational way, it’s a thought-provoking question that can lead to a real discussion.”

Another approach that works, she adds, is to take a leaf from so-called targeted job interviews and say something like, “Tell me about a time when you had a great working relationship with someone here. What made it work so well for you?” Explains Wofford, “Trying to draw out a story will give you a lot more information and insight than asking questions that call for only one-word answers.” Then follow through on what comes out of the conversation.

Your mention of giving your new coworkers time to get used to your presence is spot on, Wofford adds: “Everyone adjusts to change at a different pace. Sometimes simply recognizing and accepting that can make it easier to bide your time and focus on being great at your job.”

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Speaking of which, Wofford says that, in sessions with clients, she often hears that “work is difficult and demanding enough without also having to put all this effort into figuring out how to get along with people. The irony is, if you make the effort to understand the people you work with, the actual tasks grow easier — and you get a lot more done.”

Talkback: Have you ever stepped into a new job where you were replacing someone that others regarded as irreplaceable? How did you deal with it? Leave a comment below.