[Paper Review] Centers of symmetric cellular algebras

Yanbo Li|ArXiv.org|Nov 24, 2009

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper introduces a new ideal $ L(A) $ in the center of a symmetric cellular algebra $ A $ over an integral domain $ R $, constructed from cellular basis elements and their duals. It proves $ L(A) $ contains the Higman ideal, is independent of the symmetrizing trace, and when $ R $ is a field, its dimension is at least the number of non-isomorphic simple $ A $-modules, extending known results on centers of cellular algebras.

ABSTRACT

Let $R$ be an integral domain and $A$ a symmetric cellular algebra over $R$ with a cellular basis $\{C_{S,T}^\lam \mid \lam\inΛ, S,T\in M(\lam)\}$. We will construct an ideal $L(A)$ of the center of $A$ and prove that $L(A)$ contains the so-called Higman ideal. When $R$ is a field, we prove that the dimension of $L(A)$ is not less than the number of non-isomorphic simple $A$-modules.

Motivation & Objective

• To extend the Higman ideal to a broader class of symmetric cellular algebras.
• To construct an explicit, trace-independent ideal $ L(A) $ within the center of such algebras.
• To establish a lower bound on the dimension of $ L(A) $ when the base ring is a field.
• To provide a combinatorial framework for understanding the center of symmetric cellular algebras without relying on Weyl group structures.

Proposed method

• Define $ x_{ heta} = \sum_{S \in M(\lambda)} C_{S,T}^{\lambda} D_{S,T}^{\lambda} $ for fixed $ T \in M(\lambda) $, independent of $ T $, forming basis elements of $ L(A) $.
• Construct $ L(A) = \left\{ \sum_{\lambda \in \Lambda} r_\lambda x_\lambda \mid r_\lambda \in R \right\} $, an $ R $-submodule of the center $ Z(A) $.
• Prove $ L(A) $ is an ideal of $ Z(A) $ using properties of symmetrizing traces and dual bases.
• Show $ L(A) $ contains the Higman ideal $ H(A) = \left\{ \sum_{\lambda,S,T} C_{S,T}^{\lambda} a D_{S,T}^{\lambda} \mid a \in A \right\} $ via trace and structure arguments.
• Establish independence of $ L(A) $ from the choice of symmetrizing trace $ \tau $ using trace duality and basis invariance.
• In the semisimple case over a field $ K $, show $ \{ c_{W(\lambda)}^{-1} x_\lambda \} $ forms a complete set of primitive central idempotents in $ Z(A_K) $.

Experimental results

Research questions

• RQ1 Can the Higman ideal in the center of a symmetric cellular algebra be extended to a larger, more structured ideal?
• RQ2 Is there a canonical, trace-independent construction of a center ideal in symmetric cellular algebras?
• RQ3 What is the minimal dimension of such a center ideal when the base ring is a field?
• RQ4 Can the central idempotents of the semisimple center be explicitly constructed from the cellular basis?
• RQ5 How do Schur elements and characters relate to the new ideal $ L(A) $ in the semisimple case?

Key findings

• The ideal $ L(A) $ is an ideal of the center $ Z(A) $ and strictly contains the Higman ideal $ H(A) $, providing a larger canonical subideal.
• The construction of $ L(A) $ is independent of the choice of symmetrizing trace $ \tau $, ensuring invariance under different trace realizations.
• When $ R $ is a field, the dimension of $ L(A) $ is at least the number of non-isomorphic simple $ A $-modules, offering a lower bound on the center's size.
• In the semisimple case over a field $ K $, the elements $ c_{W(\lambda)}^{-1} x_\lambda $ form a complete set of primitive central idempotents in $ Z(A_K) $, explicitly linking the cellular structure to the center's idempotent decomposition.
• The condition $ a_\lambda c_{W(\lambda)} n_\lambda \in R $ is necessary for $ \sum_\lambda a_\lambda x_\lambda \in A $, giving a criterion for integrality of elements in the center.
• The set $ \{x_\lambda \mid \lambda \in \Lambda\} $ forms a $ K $-basis of $ Z(A_K) $, showing that $ L(A_K) $ spans the entire center in the semisimple case.