When supervisors are promoted, they should have a subtitle under their new title: resolver of conflicts, great and small, which threatens workplace cohesion, productivity, and morale. Might be a bit long for a plaque on your door, but when you are promoted into a position of authority, one of your most critical responsibilities is managing and resolving conflict. The first step is recognizing that this is now part of your job description. Next up is figuring out how to get the issue handled and your team back to work.
You have to start to figure out what is at stake for the parties involved in the conflict. What is the big issue? Until that is sorted out, it is difficult to develop a viable solution. In the language of mediation, it's all about identifying and clarifying interests.
What is really underlying the issues that crop up? It's not the details we see on the surface that are really at stake. Those are symptoms, and we want to find the root cause. It could be a case of respect, not feeling appreciated, fear of losing one's job, problems at home or with aging parents, etc. Whatever it is, identifying the underlying interests that are at stake is essential. This is what is keeping people stuck in conflict, and understanding those interests will help break them out of it.
Let's see how this looks in action: recently, I worked with a supervisor who was in charge of a team of about 12 people. Everyone under him was at the same level, although they had varying levels of experience and years on the job. Because the supervisor had to be out of the office often, he needed a "right hand man," someone to step up and take the leadership role when he was not there. A senior member of the team took on the extra responsibility (though with no extra pay or official designation).
This unofficial, second-in-command was excellent at administrative tasks, and he took over this role because the supervisor didn't have the time or the same aptitude for it. What was troublesome was the leadership aspect. The second-in-command had lost the respect of the rest of the team for things like giving himself preferential shift schedules. At least he was perceived to have done this.
The supervisor was in a fix; he needed someone he could count on to lead, and he needed these critical administrative duties completed. He also knew that he would not be viewed as much of a leader if the complaints by his team about the second-in-command were not addressed. He had practical needs but his own leadership was also at stake. The question was, "How are you going to get these practical needs met and develop a solution that demonstrates leadership?" One possibility he came up with was to have the senior member continue to handle administrative tasks and find someone else to handle the leadership responsibilities. The problem was how would the second in command handle the "demotion". This is why the supervisor was stuck.
We explored what was at stake for the senior member. He had enjoyed a position of trust and responsibility as second-in-command. He had felt valued knowing his experience and skill were recognized. But that just wasn't the case anymore. The team no longer trusted him and he knew it. He was stuck simply demanding the respect his position was meant to carry but this was a hollow claim to respect. Staying in the position would not get him what he wanted. The two decided on a way to allow the senior member to continue to handle administrative duties while working to regain trust and respect. If all went well, he could be "reinstated" as the second-in-command after a period of time.
Everyone got what they wanted: the supervisor got his administrative tasks done and the leadership role filled by someone who his team did trust. The supervisor demonstrated his own leadership to his team by responding to their concerns. The senior member continued to do what he excelled at (admin tasks), as well as having the opportunity to regain trust, which he recognized was not going to happen overnight. He actually benefited from the acknowledgment to the team that he needed to start earning back their trust. He was headed in the direction he wanted to go. The team felt that its issues were addressed and were able to move forward more cohesively.
If the supervisor had let this conflict simmer, it would have eventually boiled over and affected the team's work. He would have lost his team's respect and someone might even have been injured because of the dangerous type of work involved. Instead, figuring out what was at stake for each and developing a solution that allowed everyone's interests to be met created a more productive environment for the entire team.