# Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras

Benanti, Francesca; Boumova, Silvia; Drensky, Vesselin; K. Genov, Georgi; Koev, Plamen

Serdica Mathematical Journal (2012)

- Volume: 38, Issue: 1-3, page 137-188
- ISSN: 1310-6600

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topBenanti, Francesca, et al. "Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras." Serdica Mathematical Journal 38.1-3 (2012): 137-188. <http://eudml.org/doc/288265>.

@article{Benanti2012,

abstract = {2010 Mathematics Subject Classification: 05A15, 05E05, 05E10, 13A50, 15A72, 16R10, 16R30, 20G05Let K be a field of any characteristic. Let the formal power series f(x1, ..., xd) = ∑ αnx1^n1 ··· xd^nd = ∑ m(λ)Sλ(x1, ..., xd), αn, m(λ) ∈ K, be a symmetric function decomposed as a series of Schur functions. When f is a rational function whose denominator is a product of binomials of the form 1−x1^a1 ··· xd^ad, we use a classical combinatorial method of Elliott of 1903
further developed in the Ω-calculus (or Partition Analysis) of MacMahon in 1916 to compute the generating function X
M(f;x1, ..., xd ) = ∑ m(λ)x1^λ1 ··· xd^λd, λ = (λ1, ..., λd). M is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.The research of the first named author was partially supported by INdAM. The research of the second, third, and fourth named authors was partially supported by Grant for Bilateral Scientific Cooperation between Bulgaria and Ukraine. The research of the fifth named author was partially supported by NSF Grant DMS-1016086.},

author = {Benanti, Francesca, Boumova, Silvia, Drensky, Vesselin, K. Genov, Georgi, Koev, Plamen},

journal = {Serdica Mathematical Journal},

keywords = {Rational Symmetric Functions; MacMahon Partition Analysis; Hilbert Series; Classical Invariant Theory; Noncommutative Invariant Theory; Algebras with Polynomial Identity; Cocharacter Sequence},

language = {eng},

number = {1-3},

pages = {137-188},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras},

url = {http://eudml.org/doc/288265},

volume = {38},

year = {2012},

}

TY - JOUR

AU - Benanti, Francesca

AU - Boumova, Silvia

AU - Drensky, Vesselin

AU - K. Genov, Georgi

AU - Koev, Plamen

TI - Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras

JO - Serdica Mathematical Journal

PY - 2012

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 38

IS - 1-3

SP - 137

EP - 188

AB - 2010 Mathematics Subject Classification: 05A15, 05E05, 05E10, 13A50, 15A72, 16R10, 16R30, 20G05Let K be a field of any characteristic. Let the formal power series f(x1, ..., xd) = ∑ αnx1^n1 ··· xd^nd = ∑ m(λ)Sλ(x1, ..., xd), αn, m(λ) ∈ K, be a symmetric function decomposed as a series of Schur functions. When f is a rational function whose denominator is a product of binomials of the form 1−x1^a1 ··· xd^ad, we use a classical combinatorial method of Elliott of 1903
further developed in the Ω-calculus (or Partition Analysis) of MacMahon in 1916 to compute the generating function X
M(f;x1, ..., xd ) = ∑ m(λ)x1^λ1 ··· xd^λd, λ = (λ1, ..., λd). M is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.The research of the first named author was partially supported by INdAM. The research of the second, third, and fourth named authors was partially supported by Grant for Bilateral Scientific Cooperation between Bulgaria and Ukraine. The research of the fifth named author was partially supported by NSF Grant DMS-1016086.

LA - eng

KW - Rational Symmetric Functions; MacMahon Partition Analysis; Hilbert Series; Classical Invariant Theory; Noncommutative Invariant Theory; Algebras with Polynomial Identity; Cocharacter Sequence

UR - http://eudml.org/doc/288265

ER -

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