A Pieri Rule for Hermitian Symmetric Pairs I
نویسندگان
چکیده
Let (G,K) be a Hermitian symmetric pair and let g and k denote the corresponding complexified Lie algebras. Let g = k⊕p+⊕p− be the usual decomposition of g as a k-module. K acts on the symmetric algebra S(p−). We determine the K-structure of all K-stable ideals of the algebra. Our results resemble the Pieri rule for Young diagrams. The result implies a branching rule for a class of finite dimensional representations that appear in the work of Enright and Willenbring (preprint, 2001) and Enright and Hunziker (preprint, 2002) on Hilbert series for unitarizable highest weight modules.
منابع مشابه
A Pieri Rule for Hermitian Symmetric Pairs
Let (G, K) be a Hermitian symmetric pair and let g and k denote the corresponding complexified Lie algebras. Let g = k⊕ p+⊕ p− be the usual decomposition of g as a k-module. K acts on the symmetric algebra S(p−). We determine the K-structure of all K-stable ideals of the algebra. Our results resemble the Pieri Rule for Young diagrams. The result implies a branching rule for a class of finite di...
متن کاملA Pieri Rule for Hermitian Symmetric Pairs Ii
Let X be an irreducible Hermitian symmetric space of noncompact type and rank r. Let p ∈ X and let K be the isotropy group of p in the group of biholomorphic transformations. Let S denote the symmetric algebra in the holomorphic tangent space to X at p. Then S is multiplicity free as a representation of K and the irreducible constituents are parametrized by r-tuples, (m1, . . . ,mr) with m1 ≥ ·...
متن کاملMultiplicity-free theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs
The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known res...
متن کاملAffine Insertion and Pieri Rules for the Affine Grassmannian
We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: • Pieri rules for the Schubert bases of H∗(Gr) and H∗(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. • A new combinatorial definition for k-Schur functions, which represent the Schubert ba...
متن کاملWreath Product Symmetric Functions
We systematically study wreath product Schur functions and give a combinatorial construction using colored partitions and tableaux. The Pieri rule and the Littlewood-Richardson rule are studied. We also discuss the connection with representations of generalized symmetric groups.
متن کامل