By H. W. Turnbull
Thorough and self-contained, this penetrating examine of the idea of canonical matrices provides an in depth attention of all of the theory's imperative positive factors. subject matters comprise undemanding variations and bilinear and quadratic types; canonical aid of an identical matrices; subgroups of the crowd of an identical changes; and rational and classical canonical varieties. the ultimate chapters discover a number of equipment of canonical aid, together with these of unitary and orthogonal differences.
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Additional info for An introduction to the theory of canonical matrices, by H.W. Turnbull and A.C. Aitken
It is one-dimensional and contains the function χ(detg). Moreover the GF -modules Bs (µ1 , µ2 ) and B(µ2 , µ1 )/Bf (µ2 , µ1 ) are equivalent as are the modules B(µ1 , µ2 )/Bs (µ1 , µ2 ) and Bf (µ2 , µ1 ). We start with a simple lemma. 1 Suppose there is a non-zero function f in B(µ1 , µ2 ) invariant under right translations −1/2 by elements of NF . Then there is a quasi-character χ such that µ1 = χαF f is a multiple of χ. 0 −1 1 0 1/2 and µ2 = χαF and Since NF AF 01 −1 0 NF is an open subset of GF the function f is determined by its value at .
The proposition follows. 23. Let π be an absolutely cuspidal representation and assume the largest ideal on which ψ is trivial is OF . Then, for all characters ν, Cn (ν) = 0 if n ≥ −1. Take a character ν and choose n1 such that Cn1 (ν) = 0. Then Cn (ν) = 0 for n = n1 . If ν = ν −1 ν0−1 then, as we have seen, C(ν, t)C(ν, t−1 z0−1 ) = ν0 (−1) so that Cn (ν) = 0 for n = n1 and Cn1 (ν)Cn1 (ν) = ν0 (−1)z0n1 . 11 take n = p = n1 + 1 to obtain η(σ −1 ν, n1 +1 )η(σ −1 ν, n1 +1 )C2n1 +2 (σ) = z0n1 +1 ν0 (−1) + (| | − 1)−1 z0 Cn1 (ν)Cn1 (ν).
Chapter 1 52 −1 If µ1 µ−1 2 is not αF or αF so that ρ(µ1 , µ2 ) is irreducible we let π(µ1 , µ2 ) be any representation in the class of ρ(µ1 , µ2 ). If ρ(µ1 , µ2 ) is reducible it has two constituents one finite dimensional and one infinite dimensional. A representation in the class of the first will be called π(µ1 , µ2 ). A representation in the class of the second will be called σ(µ1 , µ2 ). Any irreducible representation which is not absolutely cuspidal is either a π(µ1 , µ2 ) or a σ(µ1 , µ2 ).