International Congress of Mathematicians

646 Alaxim NazarovWhen p = (0,0,...), this is the usual Young diagram of the partition A. Considerthe Y(gl JV )-module obtained from the Ajv(Af )-module (1.5) by pulling back throughthe homomorphism O.NM ° T Z : Y(gl N ) —t Ajv(Af). Since the central elements ofU(gl JV+M ) act in (1.5) as scalar operators, this Y(gl JV )-module is irreducible. It isdenoted by V u (z), and is called an elementary module. Its equivalence class doesnot depend on the choice of the integer M, such that A{ ^ N + M and p[ ^ M.The elementary modules are distinguished amongst all irreducible Y(gl N )-modules by the following theorem. Consider the chain of algebrasY(gl 1 )cY(gl 2 )c...cY(gl JV ). (1.7)Here for every k = 1, ... ,N — 1 we use the embedding (pi : Y(gt k ) —¥ Y(gl fc+1 ).Consider the subalgebra of Y(gl JV ) generated by the centres of all algebras in thechain (1.7), it is called the Gelfand-Zetlin subalgebra. This subalgebra is maximalcommutative in Y(gl N ); see [3] and [13]. Take any finite-dimensional module Wover the Yangian Y(gl N ).Theorem 1. Two conditions on the Y(gl N )-module W are equivalent:a) W is irreducible, and the action of the Gelfand-Zetlin subalgebra of Y(gl N )in W is semi-simple;b) W is obtained by pulling back through some automorphism (1.4) from thetensor productV U!l (zi)®...®V U!m (z m ) (1.8)of elementary Y (gl N)-modules, for some skew Young diagrams OJI, ... ,oj m and forsome complex numbers zi, ...,z m such that Zk — z% $. Z for all k fi^l.This characterization of irreducible finite-dimensional Y(gl JV )-modules withsemi-simple action of the Gelfand-Zetlin subalgebra was conjectured by Cherednik,and was proved by him [3] under certain extra conditions on the module W. In fullgenerality, Theorem 1 was proved in [14]. An irreducibility criterion for the Y(gl JV )-module (1.8) with arbitrary parameters Zi, ... ,z m was given in [15].The classification of all irreducible finite-dimensional Y(gl JV )-modules has beengiven by Drinfeld [5]. However, the general structure of these modules needs a betterunderstanding. For instance, the dimensions of these modules are not explicitlyknownin general. The tensor products (1.8) provide a wide class of irreducibleY(gljv)~ m odules, which can be constructed explicitly.1.4. The Y(gl JV )-module V u (z) has an explicit realization. It extends the classicalrealization of irreducible g I N -module V v by means of the Young symmetrizers [20].Let us use the standard graphic representation of Young diagrams on the planeR 2 with two matrix style coordinates. The first coordinate increases from top tobottom, the second coordinate increases from left to right. The element (i,j) £ UJis represented by the unit box with the bottom right corner at the point (i,j) £ R 2 •Suppose the set UJ consists of n elements. Consider the column tableau of shapeUJ. It is obtained by filling the boxes of UJ with numbers 1, ..., n consecutively bycolumnsfrom left to right, downwards in every column. Denote this tableau by 0.