Amitai Regev

Hook young diagrams with applications to combinatorics and to representations of Lie superalgebras

Definition et etude des tableaux (k,l)-semi-standards. Applications a la combinatoire et aux representations des superalgebres de Lie des diagrammes de Young a crochets

Wreath products and P.I. algebras

Abstract The representation theory of wreath products G ∼ S n is applied to study algebras satisfying polynomial identities that involve a group G of (anti)automorphisms, in the same way the representation theory of S n was applied earlier to study ordinary P.I. algebras. Some of the basic results of the ordinary case are generalized to the G -case.

CODIMENSIONS AND TRACE CODIMENSIONS OF MATRICES ARE ASYMPTOTICALLY EQUAL

The codimensionscn and the trace codimensionstn of thek×k matrices are asymptotically equal: limn→∞(tn/cn)=1. Sincetn≅q(n)·k2n whereq(x) is a known rational function, this asymptotically givescn. This has applications to the codimensions of Capelli identities.

Applications of hook Young diagrams to P.I. Algebras

Abstract (a) The multiplicities mλ in the cocharacter χn(A) of (any P.I. algebra A) are polynomially bounded, (b) A hook containing χn(A ⊗ B) is obtained from the hooks containing χn(A) and χn(B). These results are obtained by applying a theory of hook Young diagrams to P.I. algebras, and they generalize results known for algebras satisfying Capelli identities.

Algebras satisfying a Capelli identity

The sequence of cocharacters (c.c.s.) of a P.I. algebra is studied. We prove that an algebra satisfies a Capelli identity if, and only if, all the Young diagrams associated with its cocharacters are of a bounded height. This result is then applied to study the identities of certain P.I. algebras, includingFk.

PI-algebras and their cocharacters

Abstract All PI-algebras R satisfy identities of the form: f ∗ [x, y] = Σ α σ x σ(1) y 1 x σ (2)y 2 … y n − 1 x σ(n) (Theorem A.) The existence of these identities imply also that cocharacters x n ( R ) lie in a hook-shaped strip of width depending on the degree of the minimal identity of R (Theorem C). This extends a characterization of rings satisfying a Capelli identity (Theorem B).

Frobenius-Schur Functions

We introduce and study a new basis in the algebra of symmetric functions. The elements of this basis are called the Frobenius—Schur functions (FS- functions, for short).

Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities

Let A be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence c n (A) is asymptotic to a function of the form an g l n , where l ∈ N and g ∈ Z.

The representations of Sn and explicit identities for P. I. algebras

Abstract The representation theory of S n is applied to prove that for any P.I. algebra, FS n contains a certain two-sided ideal of identities, provided that n is big enough. In characteristic zero this yields Amitsurs s l k [ x ] theorem together with explicit l and k . The same methods yield an explicit identity for A ⊗ B . Modular representations of S n are used to extend those results to an arbitrary characteristic.

Codimensions of products and of intersections of verbally primeT-ideals

By Kemer’s theory [9],T idealsJ1 ∪…∪Jr andJ1 …Jr, where eachJi is verbally prime, are of fundamental importance in the theory of P.I. algebras. We calculate, approximately and asymptotically, the codimensions of suchT-ideals, thereby extending the corresponding results about matrix algebras. In all such cases, the exponential growth of the codimensions is calculated; in particular, it is always an integer.