zbMATH — the first resource for mathematics

Multigraded Hilbert series of noncommutative modules. (English) Zbl 1418.16033
Summary: In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence 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 fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.

16Z05 Computational aspects of associative rings (general theory)
16P90 Growth rate, Gelfand-Kirillov dimension
Full Text: DOI arXiv
[1] Bayer, D.; Stillman, M., Computation of Hilbert functions, J. Symbolic Comput., 14, 31-50, (1992) · Zbl 0763.13007
[2] Benanti, F.; Boumova, S.; Drensky, V.; Genov, G. K.; Koev, P., Computing with rational symmetric functions and applications to invariant theory and PI-algebras, Serdica Math. J., 38, 137-188, (2012) · Zbl 1374.13009
[3] Berele, A.; Regev, A., Applications of hook Young diagrams to P.I. algebras, J. Algebra, 82, 559-567, (1983) · Zbl 0517.16013
[4] Bigatti, A. M., Computation of Hilbert-Poincaré series, J. Pure Appl. Algebra, 119, 237-253, (1997) · Zbl 0896.13013
[5] Boumova, S.; Drensky, V., Cocharacters of polynomial identities of upper triangular matrices, J. Algebra Appl., 11, (2012), 24 pages · Zbl 1248.16020
[6] Cohn, P. M., Free ideal rings and localization in general rings, New Mathematical Monographs, vol. 3, (2006), Cambridge University Press Cambridge · Zbl 1114.16001
[7] Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H., Singular 4-1-0 — a computer algebra system for polynomial computations, (2016)
[8] de Luca, A.; Varricchio, S., Finiteness and regularity in semigroups and formal, Languages, Monographs in Theoretical Computer Science. An EATCS Series, (1999), Springer-Verlag Berlin · Zbl 0935.68056
[9] Drensky, V., Free algebras and PI-algebras. graduate course in algebra, (2000), Springer-Verlag Singapore · Zbl 0936.16001
[10] Drensky, V.; La Scala, R., Gröbner bases of ideals invariant under endomorphisms, J. Symbolic Comput., 41, 835-846, (2006) · Zbl 1125.16012
[11] Fulton, W., Young tableaux. with applications to representation theory and geometry, London Mathematical Society Student Texts, vol. 35, (1997), Cambridge University Press Cambridge · Zbl 0878.14034
[12] Giambruno, A.; Zaicev, M., Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, (2005), American Mathematical Society Providence, RI · Zbl 1105.16001
[13] Govorov, V. E., Graded algebras, Mat. Zametki, Math. Notes, 12, 1972, 552-556, (1973), (Russian), translation in · Zbl 0253.16003
[14] Krakowski, D.; Regev, A., The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc., 181, 429-438, (1973) · Zbl 0289.16015
[15] La Scala, R.; Levandovskyy, V., Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput., 44, 1374-1393, (2009) · Zbl 1186.16014
[16] La Scala, R.; Levandovskyy, V., Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput., 48, 110-131, (2013) · Zbl 1272.16026
[17] La Scala, R., Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms, Internat. J. Algebra Comput., 24, 1157-1182, (2014) · Zbl 1329.16039
[18] La Scala, R., Monomial right ideal and the Hilbert series of noncommutative modules, J. Symbolic Comput., 80, 403-415, (2017) · Zbl 1349.16095
[19] La Scala, R., Computing minimal free resolutions of right modules over noncommutative algebras, J. Algebra, 478, 458-483, (2017) · Zbl 1406.16005
[20] Latyshev, V. N., On the choice of basis in a T-ideal, Sibirsk. Mat. Zh., 4, 1122-1127, (1963), (Russian) · Zbl 0199.07501
[21] Laue, R., Symmetrica
[22] Maltsev, Yu. N., A basis for the identities of the algebra of upper triangular matrices, Algebra Logika, Algebra Logic, 10, 242-247, (1971), (Russian), translation in · Zbl 0296.16008
[23] Stembridge, J., John Stembridge’s Maple packages for symmetric functions, posets, root systems and finite Coxeter groups
[24] Ufnarovski, V. A., A growth criterion for graphs and algebras defined by words, Mat. Zametki, Math. Notes, 31, 238-241, (1982), (Russian), translation in · Zbl 0528.05060
[25] Ufnarovski, V. A., On the use of graphs for calculating the basis, growth and Hilbert series of associative algebras, Mat. Sb., Math. USSR, Sb., 68, 417-428, (1991), (Russian), translation in
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.