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Θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field. introduced θ10 for the symplectic group Sp4(F''q'') over a finite field F''q'' of order ''q'', and showed that in this case it is ''q''(''q'' – 1)2/2-dimensional. The subscript 10 in θ10 is a historical accident that has stuck: Srinivasan arbitrarily named some of the characters of Sp4(F''q'') as θ1, θ2, ..., θ13, and the tenth one in her list happens to be the cuspidal unipotent character. θ10 is the only cuspidal unipotent representation of Sp4(F''q''). It is the simplest example of a cuspidal unipotent representation of a reductive group, and also the simplest example of a degenerate cuspidal representation (one without a Whittaker model). General linear groups have no cuspidal unipotent representations and no degenerate cuspidal representations, so θ10 exhibits properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |