The Delta-filtered modules without self-extensions for the Auslander algebra of k[T]/< T^n>

With L. Hille, G. Röhrle and C.M. Ringel.
Algebras and Representation Theory 2 (1999), 295-312.

ABSTRACT. It is well-known that the Auslander algebra of any representation finite algebra is quasi-hereditary. We consider the Auslander algebra A_n of k[T]/<T^n> (here, k is a field, T a variable and n a natural number). We determine all Delta-filtered A_n-modules without self-extensions; they can be described purely combinatorially. We show, given any Delta-filtered module N, there is (up to isomorphism) a unique Delta-filtered module M without self-extensions which has the same dimension vector. In case k is an infinite field, N is a degeneration of this module M. In particular, we see that in this case, the set of Delta-filtered modules with a fixed dimension vector is the closure of an open orbit (thus irreducible). As observed in [HiRö], the problem of describing all Delta-filtered A_n-modules is the same as that of describing the conjugacy classes of elements in the unipotent radical of a parabolic subgroup P of GL(m,k) under the action of P, thus we recover Richardson's dense orbit theorem in this instance.

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