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New presentations of Specht modules of symmetric groups over fields ofcharacteristic zero have been obtained by Brauner, Friedmann, Hanlon, Stanleyand Wachs. These involve generators that are column tabloids and relations thatare symmetrizations of Garnir relations with minimal number of exchangesbetween consecutive columns or Garnir relations with maximal number ofexchanges between consecutive columns. In this paper, we examinesymmetrizations of Garnir relations and Garnir relations with any number ofexchanges. In both cases, we provide sufficient arithmetic conditions so thatthe corresponding quotient is a Specht module. In particular, in the secondcase this yields new presentations of Specht modules if the parts of theconjugate partition that correspond to maximal number of exchanges greater than1 are distinct. These results generalize the presentations mentioned above andoffer an answer to a question of Friedmann, Hanlon and Wachs. Our approach isvia representations of the general linear group.

Presentations of Schur and Specht modules in characteristic zero