The SB590 has a belt drive in what would be the chaincase on other SBs that I am (very vaguely) familiar with. This was a complete surprise to me, I thought they were all poor-man's ATCOs (right Darryl?) The clutch lever moves the idler at the left side, making it press against the V-belt. This tightens the belt. When the clutch lever is not being pulled, the belt should be loose enough to slip. Pulling the lever tightens the belt, making the engine pulley drive the cutter and roller pulleys. If the belt is too short, or the idler pulley is wrongly adjusted so it presses against the belt even when the lever is not being pulled, the mower will want to work: it will take off without you intending it to.
If you have just replaced the belt without adjusting the idler pulley, it sounds like the new belt is too short. If someone adjusted the idler pulley, that could just as easily be the problem. Is the idler touching the belt when the clutch lever is not being pulled?

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