Let F be a field of characteristic zero, and let
$UT_2$
be the algebra of
$2 \times 2$
upper triangular matrices over F. In a previous paper by Centrone and Yasumura, the authors give a description of the action of Taft’s algebras
$H_m$
on
$UT_2$
and its
$H_m$
-identities. In this paper, we give a complete description of the space of multilinear
$H_m$
-identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally prove that the variety of
$H_m$
-module algebras generated by
$UT_2$
has the Specht property, i.e., every
$T^{H_m}$
-ideal containing the
$H_m$
-identities of
$UT_2$
is finitely based.
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