5,196 reputation
829
bio website cs.ox.ac.uk/people/…
location Oxford, United Kingdom
age 51
visits member for 4 years, 5 months
seen 27 mins ago

14h
reviewed Leave Closed Maximality statements that cannot be proved using $\mathsf{ZL}$
14h
reviewed Close Boundary Value System.
14h
reviewed Close construction of nonocommutative division rings
14h
reviewed Close Solving el Gammal given D can solve DDH
14h
reviewed Close Determining odds of a slot machine given a payout value of the icon
1d
comment Is anything known about the eigenspectrum of the regular representation of the permutation group?
@QiaochuYuan - right, I missed this point.
1d
comment Is anything known about the eigenspectrum of the regular representation of the permutation group?
@QiaochuYuan - the behaviour of a permutation matrix is not determined by the order alone; it is determined by the cyclic structure of the permutation. E.g. the multiplicity of eigenvalue 1 is the number of cycles.
1d
comment Is anything known about the eigenspectrum of the regular representation of the permutation group?
a similar argument works for elements of order 3; as an eigenvalue $\lambda$ and its conjugate must occur with the same multiplicity, we see that the number of eigenvalues equal to 1 must be equal to twice the number of eigenvalues $\zeta$, for $\zeta$ a fixed primitive cubic root of unity.
1d
comment Is anything known about the eigenspectrum of the regular representation of the permutation group?
for the elements of order 2, it's trivial that the number of eigenvalues equal to 1 equals the number of eigenvalues equal to -1, as the trace of each non-identity element equals to 0.
1d
comment Is anything known about the eigenspectrum of the regular representation of the permutation group?
@Anirbit: ARupinski claims something for the regular representation (of any finite group) only; Stembridge gives formulae for the irreducible representations of $S_n$, which is a completely different story.
1d
reviewed Leave Open The free group of a group and the kernel of a canonical morphism
1d
reviewed Close How to use Integrals to calculate the expected value of two-dimensional Gaussian distribution
1d
reviewed Close How exactly do we construct the $T^2\times \mathbb{R}$ toric Calabi-Yau three-fold?
1d
comment Is anything known about the eigenspectrum of the regular representation of the permutation group?
there is no contradiction; well, I don't see immediately how to (dis)prove ARupinski's statement, but it was probably already known to Frobenuis...
1d
answered Is anything known about the eigenspectrum of the regular representation of the permutation group?
1d
reviewed Leave Open How to teach generalizing the induction hypothesis?
1d
reviewed Close Integration over Lie groups
2d
reviewed Close algebraic closedness in in residue field
2d
reviewed Close Fractal Generators and Symmetry
2d
reviewed Leave Closed Singularities in minimal surfaces