The 2 × 2 upper triangular matrix algebra and its generalized polynomial identities

Let U T 2 be the algebra of 2 × 2 upper triangular matrices over a field F of characteristic zero. Here we study the generalized polynomial identities of U T 2 , i.e., identical relations holding for U T 2 regarded as U T 2 -algebra. We determine the generator of the T U T 2 -ideal of generalized polynomial identities of U T 2 and compute the exact values of the corresponding sequence of generalized codimensions. Moreover, we give a complete description of the space of multilinear generalized identities in n variables in the language of Young diagrams through the representation theory of the symmetric group S n . Finally, we prove that, unlike the ordinary case, the generalized variety of U T 2 -algebras generated by U T 2 has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth.