Skip to main content
SpringerLink
Log in

The Ring of \(\mathcal {T}\)-covariants

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of \(\mathcal {T}\)-covariants. This ring consists of maps defined on a ring of polynomials in one variable which commute with all the translation operators. We study this ring and we show some of its meaningful features. We state an analogue of the classical Hermite reciprocity law, and recover the Hilbert series associated with a suitable double grading via the elementary theory of partitions. Together with classical covariants of binary forms other remarkable mathematical notions, such as orthogonal polynomials and cumulants, turn out to have a natural and simple interpretation in this algebraic framework. As a consequence, a Heine integral representation for the cumulants of a random variable is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138, 126–181 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brini, A.: A Glimpse of Vector Invariant Theory. The Points of View of Weyl, Rota, De Concini and Procesi, and Grosshans. In: Damiani, E., D’Antona, O., Marra, V., Palombi, F. (eds.) From Combinatorics to Philosophy. The Legacy of G.-C. Rota, pp. 25–43 (2009)

    Chapter  Google Scholar 

  3. Dolega, M., Féray, V., Śniady, P.: Explicit combinatorial interpretation of Kerov character polynomials as number of permutation factorizations. Adv. Math. 225, 81–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chiahara, T.S.: Introduction to Orthogonal Polynomials. Dover Publications Inc. (2011)

  5. Di Nardo, E., Senato, D.: An umbral setting for cumulants and factorial moments. Eur. J. Comb. 27, 394–413 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Di Nardo, E., McCullagh, P., Senato, D.: Natural statistics for spectral samples. Ann. Stat. 41, 982–1004 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Di Nardo, E., Petrullo, P., Senato, D.: Cumulants and convolutions via Abel polynomials. Eur. J. Comb. 31, 1792–1804 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grace, W., Young, A.: The Algebra of Invariants. Cambridge University Press, Cambridge (1903)

    MATH  Google Scholar 

  9. König, W.: Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385–447 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kung, J.P.S., Rota, G.-C.: The invariant theory of binary forms. Bull. Am. Math. Soc. 10(1), 27–85 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  12. Macdonald, I.G.: Symmetric Functions and Hall Polynomials (Second Edition). Oxford Science Publications (1995)

  13. MacMahon, P.A.: Seminvariants and symmetric functions. Am. J. Math. 6, 131-163, 1883–1884

  14. McCullagh, P.: Tensor Methods in Statistics. Chapman and Hall (1987)

  15. Olver, P.J.: Classical invariant theory. London Mathematical Society Student Text 44 (1999)

  16. Novak, P., Śniady, P.: What is a free cumulant?. Not. Am. Math. Soc. 58 (2011)

  17. Petrullo, P., Senato, D.: Explicit formulae for Kerov polynomials. J Algebraic Combinatorics 33, 141–151 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Petrullo, P., Senato, D., Simone, R.: Orthogonal polynomials through the invariant theory of binary forms. Adv. Math. 303, 123–150 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Roman, S., Rota, G.-C.: The umbral calculus. Adv. Math. 27(2), 95–188 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rota, G.-C.: What is Invariant Theory, Really?. In: Crapo, H., Senato, D. (eds.) Algebraic Combinatorics and Computer Science: a Tribute to Gian-Carlo Rota. Springer (2001)

  21. Rota, G.-C., Taylor, B.D.: The classical umbral calculus. SIAM J. Math. Anal. 25, 694–711 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rota, G.-C., Shen, J.: On the combinatorics of cumulants. J Comb. Theory Ser. A 91, 283–304 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stanley, R.P.: Enumerative Combinatorics, Vol. 1 (Second edition) Cambridge University Press (2011)

  24. Sylvester, J.J.: Sur la correspondence entre deux especes différentes de deux systèmes de quantités, corrélatifs et également nombreux̀. Comptes Rendus de l’Acadé,mie des Sciences de Paris 98, 779–781 (1884)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Via dell’Ateneo Lucano 10, Potenza, Italy

    P. Petrullo & D. Senato

Authors
  1. P. Petrullo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Senato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Senato.

Additional information

Presented by Anatoly Vershik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Petrullo, P., Senato, D. The Ring of \(\mathcal {T}\)-covariants. Algebr Represent Theor 21, 511–527 (2018). https://doi.org/10.1007/s10468-017-9724-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-017-9724-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation