On tensor products of path algebras of type A

Abstract

We derive a formula for the Coxeter polynomial of the s-fold tensor productof path algebras of linearly oriented quivers of Dynkin type An i −1, in terms of the weights n1, . . . , ns 2, and show that conversely the weights can be recovered from the Coxeter polynomial of the tensor product. Our results have applications in singularity theory, in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles.

Nakayama algebras and Fuchsian singularities

“…The main result of this paper is contained in Figure 1 given below (see Theorem 6.3). 342 18 [2,3,17] [2,3,18] (2, 3, 18) ⟨2, 3, 18] ⟨2, 3,19] 36 17 [2,3,16] [2,3,17] (2, 3, 17) ⟨2, 3, 17] ⟨2, 3,18] 306 16 [2,3,15] [2,3,16] (2, 3, 16) ⟨2, 3, 16] ⟨2, 3,17] 288 15 [2,3,14] [2,3,15] (2, 3, 15) ⟨2, 3, 15] ⟨2, 3,16] 90 14 [2,3,13] [2, 3, 14] (2, 3, 14) ⟨2, 3, 14] ⟨2, 3,15] 252 13 [2,3,12] [2, 3, 13] (2, 3, 13) ⟨2, 3, 13] ⟨2, 3,14] 234 12 [2,3,11] [2,3,12] (2, 3, 12) ⟨2, 3, 12] ⟨2, 3,13] 72 11 [2, 3, 10] [2, 3, 11] (2, 3, 11...…”

“…In fact, we have the following conjecture. Conjecture 5.8 holds true between Nakayama algebras of module type, of sheaf type or of triangle type, see [16] and [14].…”

Derived equivalences between one-branch extensions of "rectangles"

“…In [5], Hille-Müller have calculated the Coxeter polynomials for the tensor product of path algebras of Dynkin type A. In particular, for A(u) we have:…”