τ-tilting theory and silting theory of skew group algebra extensions

Let Λ be a finite dimensional algebra with an action by a finite group G and A:= Λ *G the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair of the skew group algebra extension Λ ⊂ A induces a poset isomorphism between the poset of G-stable support τ-tilting modules over Λ and that of (\!\!\! G)-stable support τ-tilting modules over A. We also establish a similar poset isomorphism of posets of appropriate classes of silting complexes over Λ and A. These two results generalize and unify preceding results by Huang-Zhang, Breaz-Marcus-Modoi and the second and the third authors. Moreover, we give a practical condition under which τ-tilting finiteness and silting discreteness of Λ are inherited to those of A. As applications we study τ-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support τ-tilting modules and of silting complexes over preprojective algebra Π(Lₙ) of type Lₙ.