Jacob Towber

Yetter-Drinfel'd categories associated to an arbitrary bialgebra

Abstract Various prebraided monoidal categories associated to a bialgebra over a commutative ring are studied and their relationships at various levels are examined. Generalizations of braided bialgebras are described and prebraided monoidal categories are associated with them. Three formally different braided monoidal categories can always be associated with any bialgebra over a commutative ring. These are not necessarily the same.

Quantum deformation of flag schemes and Grassmann schemes. I. A q-deformation of the shape-algebra for GL(n)

An affine algebraic group G has associated with it two (closely related) Hopf algebras: namely, its ring @[Cl of representative functions, and the universal enveloping algebra U(g) of its Lie.algebra g. The concept of “q-deformation” of a Hopf algebra leads to the notion of a “quantum group” [D2, Sect. 23 or “q-deformation of G,” by which is meant a q-deformation of one of these two Hopf algebras. (A slight abuse of terminology is customary here: The Hopf algebra resulting from such a q-deformation, being in general neither commutative nor cocommutative, is not in fact still thus associated with a Lie group or algebra.) Interest in these “quantum groups” seems first to have arisen, because of their utility in obtaining solutions to the quantum Yang-Baxter equations; an exposition of these matters may be found in [FRT]; cf. also [D2]. This line of approach, via quantum groups, to the quantum Yang-Baxter equations, first introduced (so we have been informed) by Kulish and Reshetikhin [KR], and Sklyanin [S] (Faddeev’s name should also be mentioned) led to constructions related to the quantum SL(2). Independently, Woronowicz introduced quantum groups in a different way [W]. Yu I. Manin has placed these constructions in a satisfying conceptual

Representation-functors and flag-algebras for the classical groups I

i.e., to the ring R[G/U] of regular R-valued functions on the variety G/U, where U is the unipotent radical of any Bore1 subgroup of 6. Moreover, this construction is given explicitly by generators and relations, where this presentation has the property that it is a ‘“Z-form” of the flag-algebra R[G/U], i.e., not only does R[G/lJ] have generators (yi}iel and relations on these, given by the presentation, with the coefficients of these relations in Z, but also these relations generate, over Z, all relations on the (y,> with coefficients in Z. The algebra-valued functor A + is naturally graded, the gr~ding~mono~ consisting of partitions (i.e., finite unordered sequences of integers, repetitions permitted), with the monoid-operation union of partitions. Each “graded piece” Aa is a functor from R-modules to R-modules, with a presentation, given by the above construction, which is a “Z-form,” in the above sense, of a suitable irreducible representation of Gk(n); all nonzero irreducible polynomial representations of GL(M) arise exactly once in this way. These functors A’, and their analogs for the other classical groups to be described below, are the “representation-functors” in the title of [6] and of the present article; the Aa have been called “Schur functors” and ‘“shape functors”; the authors believe that [S] is the first appearance of these AE in the literature. They have recently proved useful tools in certain algebraic applications involving free resolutions. In [6], the present authors extended this work to the groups SO(21-t I) 265 0021-8693185

Reversion of power series and the extended Raney coefficients

3.00

Tensor operators III, some fundamental tensor operator identities

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients. Let F1, . . . , Fn be polynomials in variables x1, . . . , xn with complex coefficients, where n > 2. Suppose, for each i, Fi = xi + higher degree terms and the Jacobian determinant of F1, . . . , Fn is equal to 1. Then the Jacobian Conjecture [1], [9] asserts, in this case, that x1, . . . , xn are also polynomials in F1, . . . , Fn with complex coefficients. This long-standing conjecture has not been solved even for n = 2. Since it can be proved that x1, . . . , xn are (formal) power series in F1, . . . , Fn with complex coefficients, the Jacobian Conjecture asserts that these power series are really polynomials. This provides the motivation for this paper. Let F1, . . . , Fn be power series in variables x1, . . . , xn of the form Fi = xi + higher degree terms with indeterminate coefficients for each i. It is known (e.g. [2, Chapter III, Section 4.4, Proposition 5, p. 219]) that F = (F1, . . . , Fn) has a (unique) compositional inverse, i.e., there exists G = (G1, . . . , Gn) where each Gi is a power series in variables x1, . . . , xn such that F ◦ G = 1 and G ◦ F = 1, or equivalently, Fi(G1, . . . , Gn) = xi and Gi(F1, . . . , Fn) = xi for all i. There are various methods in the literature to find the coefficients of Gi. In this paper we shall present two new ones. Since each coefficient of Gi is a polynomial in the indeterminate coefficients of F1, . . . , Fn, it is enough to find the coefficients of these polynomials. We will refer to these coefficients as the (extended) Raney coefficients. In the first method, generating functions in infinitely many variables are used to show that each Raney coefficient has a combinatorial interpretation as the number of colored trees in a certain collection (Theorems 2.4 and 2.5). In Received by the editors April 4, 1994. 1991 Mathematics Subject Classification. Primary 13F25, 05A15, 05C05, 13P99; Secondary 32A05.

A Combinatorial Problem in the Representation Theory of SL(n)

Let s≥2 and let α=<a1,a2,...,as≳, where s, ai ∈ Z and a1≥a2≥...≥as≥1. Let 1≤t≤s. The method of undetermined coefficients was used to investigate natural transformations Ω:Λα → Λa1 ⊗ Λa2 ⊗... ⊗ Λas and Ωt:Λα → Λ<a1,...,at,...,as≳ ⊗ Λat. Necessary and sufficient conditions were found on the coefficients defining Ω and sufficient conditions on the coefficients defining Ωt. Explicit formulas for Ω1 and Ωs are given herein.

Two new functors from modules to algebras

Abstract. A combinatorial conjecture phrased in the language of histograms is presented. The origin of the conjecture in the representation theory of the group SL(n) is explained.