Hilbert series of noncommutative algebras, regular languages and symmetric functions

Seminario del Dipartimento di Matematica
20 febbraio 2024
Orario di inizio 
PovoZero - Via Sommarive 14, Povo (Trento)
Aula Seminari "-1"
Organizzato da: 
Dipartimento di Matematica
Comunità universitaria
Comunità studentesca UniTrento
Ingresso libero
Dott. Federico Pintore
Staff Dipartimento di Matematica
Roberto La Scala (Università di Bari)


In this talk we present a method for computing the Hilbert series of multigraded (or graded) right modules over the free associative algebra. In particular, we can compute such series for noncommutative (associative) algebras. Using results from the theory of regular languages, we provide conditions under which these methods run for a finite number of steps, ensuring that the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional
algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras
whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group.
In this case, such series are symmetric functions, allowing for their decomposition into Schur functions that correspond to the decomposition of the algebra into its irreducible submodules. 
The proposed algorithms have been fully implemented within the computer algebra system Singular (University of Kaiserslautern).