Character table

Summary

In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables. Definition and example The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by

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Computing the performance of unitary space-time group codes from their character table

Mohammad Amin Shokrollahi

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment. Their success, however, depends on the design of cedes that achieve these promises. It is well known that unitary matrices can be used to design differentially modulated space- time codes. These codes have a particularly efficient description if they form a finite group under matrix multiplication. We show how to compute the parameters of such groups crucial for their use as space-time codes, using only the character table of the group. Since character tables for many groups are known and tabulated, this method could be used to quickly test, for a given group, which of its irreducible representations can be used to design good unitary space-time codes. We demonstrate our method by computing the eigenvalues of all the irreducible representations of the special linear group

SL_2(F_q)

over a finite prime field

F_q

of odd characteristic, and study in detail the performance of a particular eight-dimensional representation of

SL_2(F_17)

2002

Character table

Group characters and unitary space-time codes

Computing the performance of unitary space-time group codes from their character table

Group characters and unitary space-time codes

Computing the performance of unitary space-time group codes from their character table