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

Let UT 2 be the algebra of 2x2 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 the generator 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 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).