Toward higher dimensional
Auslander-Reiten theory
Osamu Iyama
Let G be a finite subgroup of SLd (k) (char k = 0), S := k[[x1 , .., xd ]], S G an invariant
subring of G and S ∗ G the skew group ring.
For the case d = 2, it is well-known that S G is representation-finite, i.e. S G has only
finitely many isoclasses of indecomposable maximal Cohen-Macaulay modules. Morestrongly, Auslander found that the McKay quiver of G coincides with the AuslanderReiten quiver of S G .
For the case d > 2, S G is never representation-finite. The aim of this talk is to study
the category CMS G of maximal Cohen-Macaulay S G -modules from the viewpoint of‘higher dimensional’ Auslander-Reiten theory.
Let us recall a classical theorem of Auslander in the representation theory of finitedimensional algebras: There exists a bijection between Morita-equivalence classes ofrepresentation-finite finite-dimensional algebras and those of finite-dimensional algebras with global dimension at most two and dominant dimension at least two. This
theorem can be regarded as a starting point of Auslander-Reiten theory historically.
For a non-negative integer n, we introduce the concept of maximal n-orthogonal subcategory of modΛ (resp. CMΛ) for a finite-dimensional algebra (resp. Cohen- Macaulayring) Λ. It can be shown that there exists a bijection between equivalence classes ofmaximal (n − 2)- orthogonal subcategories and Morita-equivalence classes of finitedimensional algebras with global dimension at most n and dominant dimension atleast n.
Thus it would be natural to study ‘higher dimensional’ Auslander-Reiten theory on
maximal n-orthogonal subcategories: We can obtain ‘higher’ Auslander-Reiten duality
theorem and the existence theorem of ‘higher’ Auslander-Reiten sequences. Moreover, CMS G has a ‘canonical’ maximal (d − 2)- orthogonal subcategory addS, whose
Auslander-Reiten quiver coincides with the McKay quiver of G. But CMS G has manyother maximal (d − 2)-orthogonal subcategoires.
In this talk, we will study how to get all maximal (d − 2)-orthogonal subcategories
of CMS G . Thanks to ‘(d − 1)- Calabi-Yau’ property of CMS G , we can show that, using‘higher’ Auslander-Reiten sequences in a maximal orthogonal subcategory, one can getother maximal orthogonal subcategories [joint work with Y. Yoshino]. This process can1

be regarded as an analogy of mutation, which is well-known in representation theoryof algebras due to Rudakov (vector bundles on Pn ), Riedtmann-Schofield (classicaltilting modules), Buan-Marsh-Reineke-Reiten-Todorov (cluster categories) and GeissLeclerc-Shr´’oer (preprojective algebras). It is natural to ask whether one can get allmaximal (d − 2)-orthogonal subcategories of CMS G by successive mutations. In thetalk of Yoshino, this is shown to be true for the case G = hdiag(ω, ω, ω)i (ω 3 = 1).
We also discuss the relationship among maximal (d − 2)- orthogonal subcategories
of CMS G and the following objects [joint work with I. Reiten].
(i) Van den Bergh’s non-commutative crepant resolutions of S G ,(ii) tilting S ∗ G-modules,(iii) Fomin-Zelevinsky mutation.
Especially, we will show that certain non-commutative analogy of Bondal-Orlov
conjecture due to Van den Bergh on derived equivalences of crepant resolutions is true
for three dimensional isolated singularities.

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