The United Auto Workers on Wednesday told the New School that a majority of its teaching and research assistants have requested that UAW be recognized as their union. If the New School does not recognize the union, it said, it might ask the National Labor Relations Board to hold an election. That might set off a review of NLRB policy and create a case under which the board could reverse a ruling that has denied collective bargaining rights to teaching assistants at private universities. The request to the New School came just days after a similar request to Columbia University. A spokeswoman for the New School said that it was reviewing the request.