What muscles are used in fixed seat rowing?

A very short answer to this is that even in fixed seat rowing you will be performing a full body exercise.

Everyone will vary slightly in the way the take a stroke, but generally it will be a little something like this: 1) you lift your hands to place the oars into the water 2) you swing your body back 3) pull your hands into your body 4) you tap the oar handles down to lift the oar out of the water 5) extend your arms away from your body 5) swing your body forward to take the next stroke

The main action is the body swing which will use the muscles in your back and also your glutes. It is inevitable that you will be bending your legs a little as you swing forward and therefore using your leg muscles a little at the beginning of the stroke. Secondly, the muscles in your arms (biceps mainly) are used when you pull the handles right in to your body. Then, to tap/push the handles down toward your lap, you use your triceps and some pectoral muscles, and then the same to extend the handles away from your body. To swing the body forward, the back muscles are again used. Then some shoulder, back and arm muscles are used to lift the handle up to place the oars back into the water. Also throughout the stroke you are using your core/abdominal muscles to sit up and maintain some posture.

Arm muscles, Einstein.