Representation-finite Birkhoff type problems for nilpotent linear operators

Abstract

The paper is an addition to our article Simson (2015) [37]. Given a field K, an integer m≥1, and a finite poset I≡(I,⪯) with a unique maximal element ⁎, we study the category Mon(I,Fm) of mono-representations U=(Uj)j∈I of I over the Frobenius K-algebra Fm:=K[t]/(tm) of K-dimension m<∞. We view U as K-vector spaces U⁎, with an m-nilpotent K-linear operator t:U⁎→U⁎, together with t-invariant subspaces Ui⊆Uj⊆U⁎, for all i⪯j⪯⁎ in I. The problem of when the Krull–Schmidt K-category Mon(I,Fm) is of wild (resp. tame, finite) representation type is called a representation-wild (resp.-tame, -finite) Birkhoff type problem for m-nilpotent operators. In case when the field K is algebraically closed, we give in our previous paper a complete solution of the problem by describing all pairs (I,m), with m≥2, such that category Mon(I,Fm) is representation-tame and representation-infinite. Moreover, the tame-wild dichotomy for the category Mon(I,Fm) is proved there.In the present paper all representation-finite Birkhoff type problems are explicitly described by means of the pairs (I,m). In particular case when I=Ia,b is the union of two incomparable chains I′ and I″ of length |I′|=a−1≥1 and |I″|=b−1≥1, with I′∩I″={⁎}, we study a functorial connection vect-X(p)→Mon(Ia,b,Fm), where vect-X(p) is the vector bundle subcategory of the category coh-X(p) of coherent sheaves over the weighted projective line X(p), for the weight triple p=(a,b,m), with a,b,m≥2, applied by Kussin–Lenzing–Meltzer (2013) [18], in relation with the hypersurface singularity f=x1a+x2b+x3m. In this case we show that Mon(Ia,b,Fm) is representation-finite (resp. representation-tame, representation-wild) if and only if the orbifold Euler characteristic χ(a,b,m)=1a+1b+1m−1 of X(a,b,m) is positive (resp. non-negative, negative).