# The Ring of T $\mathcal {T}$ -covariants

The Ring of T $\mathcal {T}$ -covariants Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of T $\mathcal {T}$ -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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebras and Representation Theory Springer Journals

# The Ring of T $\mathcal {T}$ -covariants

, Volume 21 (3) – Aug 4, 2017
17 pages

/lp/springer_journal/the-ring-of-t-mathcal-t-covariants-6GBKNEJhuN
Publisher
Springer Netherlands
Subject
Mathematics; Commutative Rings and Algebras; Associative Rings and Algebras; Non-associative Rings and Algebras
ISSN
1386-923X
eISSN
1572-9079
D.O.I.
10.1007/s10468-017-9724-x
Publisher site
See Article on Publisher Site

### Abstract

Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of T $\mathcal {T}$ -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.

### Journal

Algebras and Representation TheorySpringer Journals

Published: Aug 4, 2017

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