Volume 61 A, number 4 PHYSICS LETTER S 16 May 1977 EXPONENTIAL OF GEL'FAND LATTICES AND IRREDUCIBLE REPRESENTATIONS OF U(n) J.P. GAZEAU and M.C1. DUMONT-LEPAGE Laboratoire de Chimie Physique de l'Universitd de Paris VI, rue Pierre et Marie Curie, l l, F-75005 Paris, France and A. RONVEAUX Ddpartement de Physique, Facultds Universitaires Notre-Dame de la Paix, Rue de Bruxelles, 61, B-5000 Namur, Belgium Received 29 December 1976 A finite difference equation defines the exponential of a square tableau, extension of the usual Gel'fand pattern. The application to the group U(n) gives explicitly the Gel'fand states for n = 4 and the matrix elements for n = 3. 1. Although a large amount of literature about Gel'land states exist [1-4], explicit expressions for these states or for matrix elements of irreducible rep- resentations of U(n) are still unknown for an arbi- trary n. The well known lowering procedure starting from maximal states are usually described [2, 5-9], but that construction becomes inextricable when n is larger than three [10]. We propose in this letter a combinatorial approach [11 ] completely different from the classical approach (Lie algebra, boson calculus [12, 13], tensor product, ...). Defining a Gel'fand Lattice related to the usual Gel'fand pattern, the solution of a finite difference equation gives systematically the Gel'fand states for n = 3 and 4. Furthermore, multiplication theorems give us the explicit matrix elements of U(3). Next sec- tions describe the)ssential definitions and the results which can probably be applied to the larger group GL (n, C). 2. Let us consider the following tableau closely re- lated to the usual Gel'land pattern, and called a "Gel'fand Lattice" (G.L.): t/ 1 ~ .,. k" \ m 1 T = mn m~ ,/ / j ~ :, m n gl The n 2 integers rn ! 1 ~ i,j <~ n, satisfy the between- i' ness relations for all i,j: 0~-~m{ +1 <~.m{~r/l{ 1' (1) The left and right triangular patterns will be denoted M< and M>, and the central column [m] n' With this shortened notations, a G.L. can be written r = M< Ira] n M>. (2) The following operations, stable with respect to the betweenness relations, can be defined between two G.L with same dimensions (m/. " E T, m Z" C T') • l f • i) addition T ® T = {m~ + roT} . ii) scalar multiplication pT = {vm~} (v non negative integer). If m~ - m'. / ~> 0 satisfy the betweenness relations, a l l difference can also be defined: T ® T' = {rr4 - mT}. The (2nn) Gel'fand lattices, called Binary Gel'fand Lattices (B.G.L.) are essential in this theory T=O-{rn{=O Vi, *} T = l =- {rn{ = l Vi,i} [m~:=l( i<<'l'j<~i 1 +//-i+1 t(3) T=lJa...Jl =, [j<.l,i~j-l+il_/+l) il .-.q [m~Z = 0 otherwise or explicitly 211