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Title: Sign Hibi Cones and Pieri Algebras for the General Linear Groups
Authors: WANG YI
Keywords: Sign Hibi Cones, Anti-row Iterated Pieri Algebras, Standard monomial basis, Sagbi basis, Howe correspondence
Issue Date: 10-Jun-2013
Citation: WANG YI (2013-06-10). Sign Hibi Cones and Pieri Algebras for the General Linear Groups. ScholarBank@NUS Repository.
Abstract: Let \Gamma be a finite poset. The set of all order preserving functions from \Gamma to non-negative integers forms a semigroup, and is called a Hibi cone. It has a simple structure and has been used to describe the structure of some algebras of interest in representation theory. We now consider the set of all order preserving functions from \Gamma to integers. It is also a semigroup and we call it a sign Hibi cone. We will develop the structure theory for the sign Hibi cones and some of their subsemigroups \Omega. Next, we construct an algebra A_{n,k,l} whose structure encodes the decomposition of tensor products of GLn representations of the form $\rho\otimes S^{a_1}(C^{n*})\otimes ...\otimes S^{a_l}(C^{n*})$ where \rho is a polynomial representation of GLn and S^{a_i}(C^{n*}) is the a_i-th symmetric power of C^{n*}, the dual of the standard representation of GLn on C^n. We call A_{n,k,l} an anti-row Pieri algebra for GLn. We shall construct a basis for A_{n,k,l} indexed by the elements of \Omega. We further show that this basis contains all the standard monomials on a set of generators.
Appears in Collections:Ph.D Theses (Open)

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