Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Standard character condition for table algebras. | Standard character condition for table algebras Amir Rahnamai Barghi Javad Bagherian Department of Mathematics Department of Mathematics K. N. Toosi University of Technology University of Isfahan 16315-1618 Tehran-Iran 81746-73441 Isfahan Iran rahnama@ bagherian@ Submitted Sep 23 2009 Accepted Jul 26 2010 Published Aug 9 2010 Mathematics Subject Classification 20C99 16G30 05E30 Abstract It is well known that the complex adjacency algebra A of an association scheme has a specific module namely the standard module that contains the regular module of A as a submodule. The character afforded by the standard module is called the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the standard character condition if it admits the standard character. Among other results we acquire a necessary and sufficient condition for a table algebra to originate from an association scheme. Finally we prove that given a C-algebra admits the standard character and its all degrees are integers if and only if so its dual. 1 Introduction A table or equivalently C-algebra with nonnegative structure constants was introduced by 2 . It is easy to see that the complex adjacency algebra of an association scheme or homogeneous coherent configuration is an integral table algebra. On the other hand the adjacency algebra of an association scheme has a special module namely the standard module that contains the regular module as a submodule. The character afforded by the standard module is called the standard character see 8 . This leads us to generalize the concept of standard character from adjacency algebras to table algebras. As an application of this generalization we provide a necessary and sufficient condition for a table algebra to originate from an association scheme see Theorem . The paper is organized as follows. In Section 2 we recall the concept of C-algebras and table
