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 induced by 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 (modG)-stable support τ-tilting modules over A. We also establish a similar poset isomorphism between posets of appropriate classes of silting complexes over Λ and A. These two results generalize and unify the preceding results by Zhang–Huang, 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 by 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(𝕃 n ) of type 𝕃 n .