Abstract
Consider the general linear group GL(M) over the complex field. The irreducible rational representations of the group GL(M) can be labeled by the pairs of partitions mu = (mu(1),mu(2)...) and (μ) over tilde = ((μ) over tilde (1),(μ) over tilde (2),...) such that the total number of non-zero parts of mu and (μ) over tilde does not exceed M. Let V-mu(μ) over tilde be the irreducible representation corresponding to such a pair.
Regard the direct product GL(N) x GL(M) as a subgroup of GL(N+M) . Take any irreducible rational representation V-lambda($λ) over tilde of GL(N+M). The vector space Vlambda(lambda<(λ)over) (mu(μ) over tilde)(tilde>) = Hom(GLM) (V-mu(μ) over tilde, V-lambda(λ) over tilde comes with a natural action of the group GL(N). Put n = lambda1-mu(1) + lambda(2) - mu(2) + ... and (n) over tilde = (λ) over tilde (1) - (μ) over tilde (1) + (λ) over tilde (2) - (μ) over tilde (2) + .... For any pair of standard Young tableaux Omega and (&UOmega;) over tilde of skew shapes lambda/mu and lambda/(m) over tilde respectively, we give a realization of V-lambda(λ) over tilde(mu(μ) over tilde) as a subspace in the tensor product W-n (n) over tilde of n copies of defining representation C-N of GL(N), and of (n) over tilde copies of the contragredient representation (C-N)(*). This subspace is determined as the image of a certain linear operator F-Omega(&UOmega;) over tilde on W-n (n) over tilde. We introduce this operator by an explicit multiplicative formula.
When M=0 and V-lambda(λ) over tilde(mu(μ) over tilde) is an irreducible representation of GL(N), we recover the known realization of V-lambda(λ) over tilde as a certain subspace in the space of all traceless tensors in W-n (n) over tilde. Then the operator F-Omega(&UOmega;) over tilde may be regarded as the rational analogue of the Young symmetrizer, corresponding to the tableau Omega of shape lambda . Even when M=0, our formula for F-Omega(&UOmega;) over tilde is new.
Our results are applications of the representation theory of the Yangian Y (gl(N)) of the Lie algebra gl(N). In particular, F-Omega(&UOmega;) over tilde is an intertwining operator between certain representations of the algebra Y (gl(N)) on W-n (n) over tilde. We also introduce the notion of a rational representation of the Yangian Y (gl(N)). As a representation of Y (gl(N)), the image of F-Omega(&UOmega;) over tilde is rational and irreducible.
| Original language | English |
|---|---|
| Pages (from-to) | 21-63 |
| Number of pages | 42 |
| Journal | Mathematische Zeitschrift |
| Volume | 247 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - May 2004 |
Keywords
- TENSOR-PRODUCTS