Counting equivalence classes of irreducible representations.

*(English)*Zbl 0987.16009Let \(k\) 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)].

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 |