The $2\times 2$ upper triangular matrix algebra and its generalized polynomial identities

Let $UT_2$ be the algebra of $2\times 2$ upper triangular matrices over a field $F$ of characteristic zero. Here we study the generalized polynomial identities of $UT_2$, i.e., identical relations holding for $UT_2$ regarded as $UT_2$-algebra. We determine a set of two generators of the $T_{UT_2}$-ideal of generalized polynomial identities of $UT_2$ 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 in the ordinary case, the generalized variety of $UT_2$-algebras generated by $UT_2$ has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth.