Many papers in random matrix theory make passing references to duality between eigenvalue statistics of the GOE and GSE, for which the most concrete reference I can find is <https://arxiv.org/pdf/math-ph/0206011.pdf> (page 4 has the relevant discussion). Though I'm somewhat unclear on the actual matrix integral specifics of this paper, they also note that the GUE is self-dual by their definition.


In Bott periodicity, we see something that at least superficially looks very similar, namely the isomorphisms 
$$\pi\_k(U) \cong \pi\_{k+2}(U)$$
$$\pi\_k(O) \cong \pi\_{k+4}(Sp)$$
$$\pi\_k(Sp) \cong \pi\_{k+4}(O)$$
of homotopy groups of the infinite classical groups. 


Question: is there any known relation between these two things? (I'd also appreciate any independent insight on the GOE/GSE and GUE/GUE duality). 



GOE/GSE duality and Bott periodicity
The entire set of correspondences can be read off from this table:


![](https://ilorentz.org/beenakker/MO/Botttable.png)


Listed are the 10 symmetric spaces and for each space in the left column the dual space is shown in the right column, as explained in [On duality and negative dimensions in the theory of Lie groups and symmetric spaces](https://arxiv.org/abs/1011.0151), by Ruben L. Mkrtchyan, Alexander P. Veselov (2010). The Bott periodicity modulo 2 in complex K-theory is indicated by the numbers in blue, and the Bott periodicity modulo 8 in real K-theory is indicated by the numbers in red.


The connection between the symmetric spaces $X$ and the Gaussian ensembles of random Hermitian matrices $H$ is that $e^{iH}$ is in $X$ (uniformly distributed in the limit of infinite matrix size). I have indicated the name of the random-matrix ensemble in green. In addition to the three Wigner-Dyson ensembles GOE, GUE, GSE there appear counterparts with a chiral symmetry (meaning that the Hamiltonian is block-offdiagonal). The four ensembles labeled "AZ" are the Altand-Zirnbauer ensembles referred to by John Mangual. (They describe the spectral statistics in superconductors.)


The dualities in the OP for the GUE, GOE and GSE can be read off directly from this table; in addition we find dualities between the seven remaining symmetric spaces and between their corresponding (three chiral and four AZ) random-matrix ensembles.




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`shameless plug`
If you're interested in the physics behind the AZ ensembles and how this relates to topological superconductivity/Majorana fermions, etc. you might find this [review](http://arxiv.org/abs/1407.2131) useful.


