Timeline for Nicest coset representatives of the symplectic group in the general linear group

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Dec 12, 2009 at 19:36	comment	added	David Bar Moshe		I meant the circular symplectic ensemble (not the circular clasical ensemble). Anyway, I hope that the above remark clarified the matter.
Dec 12, 2009 at 19:27	comment	added	David Bar Moshe		Sorry that I haven't given more detail. I meant quaternionic matrices where each element is a complex quaternion: a 1 + b i + c j + d k , where a, b, c, d are complex numbers. Now, this parameterization is not new, a similar parameterization is used in the circular classical ensemble, with an additional constraint of unitarity. There, the coset representaties are self dual quaternionic matrices with an additional unitarity constraint.
Dec 12, 2009 at 16:31	comment	added	José Figueroa-O'Farrill		This is not correct, I'm afraid. The $n\times n$ quaternionic matrices are in one-to-one correspondence with $\mathbb{H}$-linear endomorphisms of $\mathbb{H}^n$, whereas $2n \times 2n$ complex matrices are in one-to-one correspondence with $\mathbb{C}$-linear endomorphisms of $\mathbb{C}^{2n}$. Although as complex vector spaces $\mathbb{H}^n \cong \mathbb{C}^{2n}$, $\mathbb{H}$-linearity is a strong condition. Consider $n=1$. A $1\times 1$ quaternionic matrix is a quaternion, which is 2-dimensional over $\mathbb{C}$, whereas a $2\times 2$ complex matrix is 4-dimsensional over $\mathbb{C}$.
Dec 12, 2009 at 15:46	history	answered	David Bar Moshe	CC BY-SA 2.5