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Counting equivalence classes of irreducible representations. (English) Zbl 0987.16009
Let $$k$$ be a computable field of characteristic 0 and $$R$$ a finitely presented $$k$$-algebra. The article describes an algorithm which decides whether given a natural number $$n$$ there exist only finitely many (up to isomorphism) irreducible $$n$$-dimensional representations of $$R$$. By a representation it is meant an algebra homomorphism from $$R$$ into the $$n\times n$$-matrix algebra $$M_n(\overline k)$$ with coefficients in the algebraic closure of $$k$$. Irreducibility as well as isomorphisms are also defined over $$\overline k$$.
The algorithm uses in particular a variant of the subring membership test [see T. Becker, V. Weispfenning, Gröbner bases: a computational approach to commutative algebra, Graduate Texts in Mathematics 141 (1993; Zbl 0772.13010)]. The author sketches also a method of calculating the number of isomorphism classes of irreducible $$n$$-dimensional representations of $$R$$ in case this number is finite and $$k[t]$$ is equipped with a factoring algorithm.
Note that in a previous paper [J. Symb. Comput. 32, No. 3, 255-262 (2001; see the preceding review Zbl 0987.16008)] the author has given a procedure to verify whether $$R$$ has an $$n$$-dimensional irreducible representation for a given $$n$$. Moreover, the existence of a nonzero finite dimensional representation of $$R$$ is a Markov property and cannot be algorithmically verified in general by the result of L. A. Bokut’ [Algebra Logika 9, 137-144 (1970; Zbl 0216.01001)].
Reviewer: S.Kasjan (Toruń)

##### MSC:
 16G60 Representation type (finite, tame, wild, etc.) of associative algebras 16Z05 Computational aspects of associative rings (general theory) 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 68W30 Symbolic computation and algebraic computation
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