RICERCA ITALIANA     

Polynomial identities and combinatorial methods

Università degli Studi di Palermo

Abstract

The primary objective of the research project is the study of the polynomial identities satisfied by an algebra over a field of characteristic zero via combinatorial methods pertaining to the representation theory of the symmetric and general linear groups, to algebraic combinatorics, to invariant theory. This approach has been very fruitful in the past, and is based on the theory of varieties developed by Kemer. In this theory a fundamental role is played by the superalgebras and their identities. This approach was introduced and developed mainly by Berele, Drensky, Formanek, Kemer, Procesi, Razmyslov and Regev.
We plan to develop further the full strength of the superalgebraic version of the Capelli method of "auxiliary variables", in order to find decomposition results for some classes of "plethystic algebras"; we also plan to contribute to the "constructive invariant theory" of classical groups, with main reference to the extension of the method of transvectants, to the extension of the symbolic method to the ring of polynomial functions on arbitrary Schur/Weyl modules and to the representation theory of finite reflection groups on polynomial algebras.
This program will also involve detailed studies of specific combinatorial structures as well as the related Hopf algebras; among them, we mention the RKS-correspondence, permutation (involution) statistics and the theory of non crossing partitions.
In the theory of >>>

Principal Investigator

Antonino Giambruno Università degli Studi di PALERMO

Research Objectives

The final objectives of this research project are the following.

Computation of the exponent of a variety: to extend the classification of the varieties determined by Capelli polynomials and Amitsur polynomials to the minimal varieties. In this setting we shall try to find a closer relation between a given variety and the minimal varieties with the same exponent.
Classification of the varieties of polynomial growth: to extend the recently determined classification of the varieties of linear growth and of the subvarieties of the variety generated by the Grassmann algebra or the algebra of 2 x 2 upper triangular matrices. We shall also work on the conjecture stating that that from a certain value of the polynomial growth (>6?), the list of algebras characterizing the varieties is no more finite.
Growth of the Lie codimensions and of the proper codimensions: to determine if the limit of the sequences {c_n^p(A)^(1/n)} and {c_n^L(A)^(1/n)} exist and are integers. We plan to solve this problem for finitely generated algebras and in the general case for the proper codimensions.
Nonassociative algebras and exponential growth: to study the finite dimensional nonassociative algebras and to try to classify their exponential growth (is it an integer?). For simple Jordan algebras does the exponent exist and equals the dimension of the algebra?
Superalgebras and algebras with involution: to compare the invariants determined through the >>>

First Results

Taking into account the final targets of the research project, the expected results represent a significant contribution to the combinatorial theory of algebras with polynomial identities and to various aspects of the algebraic combinatorics and its applications. The expected results can be described as follows.

1) In the asymptotic computation of the codimension sequences of an associative algebra we shall try to obtain further information on the varieties relatyed to the exponent. Given a T-ideal of the free algebra by generators, we shall try to determine the exponent of the corresponding variety in some significant cases. Since an asymptotic equality between the sequence of codimensions of a variety of such type with some special generators (standard identity, Capelli identity, Amitsur identity) and the codimension sequence of an algebra of matrices was determined earlier, we intend to extend this classification to the minimal varieties.

2) Starting from the explicit classification, up to PI-equivalence, of the algebras whose sequence of codimensions is linearly bounded, and from the classification of the subvarieties of the variety generated by the Grassmann algebra or the algebra of 2 x 2 upper triangular matrices, we expect to extend these results to higher growth. In particular we shall work at the conjecture stating that for a suitable value of the growth (>6) the list of algebras characterizing the corresponding varieties is no >>>

Timescale

24 months

National and international background

The use of the representation theory of the symmetric group for studying the T-ideals of the free associative algebra was systematically made since the early 70's; this method was developed in characteristic zero by Regev, Kemer, Procesi, Razmyslov, Amitsur, Berele, Drensky and others.
The theory of Lie algebras with polynomial identities was developed by the Moscow school by Bahturin, Zaicev, Mishchenko, Petrogradski, Razmyslov and others (Y. Bahturin, A. A. Mikhalev, V. Petrogradsky, M. Zaicev, Infinite Dimesional Lie Superalgebras, Walter de Gruyter, Berlin, 1992).

The study of the asymptotic behavior of the sequence of codimensions for associative algebras was started by Regev by proving that for a PI-algebra the sequence of codimensions is exponentially bounded (Existence of identities in A x B, Israel J. Math. 11 (1972), 131-152). The theorem asserting that every proper variety has integral exponential growth and its explicit computation is due to Giambruno and Zaicev (On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145-155; Exponential codimension growth of PI-algebras: an exact estimate, Adv. Math., 142 (1999), 221-243). The computation of the exponent and the methods introduced have allowed to compute the exponential growth of polynomials. This study was started by Berele and Regev in [ Exponential growth for codimensions of some p.i. algebras. J. Algebra 241 (2001), 118-145 ]. The study of the >>>