bio  website  math.berkeley.edu/~qchu 

location  Berkeley, CA  
age  24  
visits  member for  5 years, 6 months 
seen  47 mins ago  
stats  profile views  59,983 
I am a thirdyear graduate student at UC Berkeley. I'm interested in the interplay between homotopy theory, quantum field theory, manifold topology, and higher category theory.
1d

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Is anything known about the eigenspectrum of the regular representation of the permutation group?
@Dima: the behavior of the permutation matrix describing multiplication by $g$ is completely determined by the order of $g$; the only possible cycle length is the order of $g$. 
2d

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Is anything known about the eigenspectrum of the regular representation of the permutation group?
@user6818: thinking about the irrep decomposition actually makes this harder. You already know what $R(g)$ looks like in the regular representation: it's the permutation matrix of the permutation corresponding to multiplication by $g$. The behavior of this permutation matrix is completely determined by the order of $g$ and that's all there is to it. A similarly easy result which again does not require you to know anything about the irreps of $G$ is that the trace of $R(g)$ is $0$ if $g$ is not the identity. 
2d

awarded  Nice Answer 
Apr 23 
awarded  Nice Question 
Apr 23 
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When does an irreducible unitary real representation remain irreducible after complexifying it?
I think "unitary real representation" just means it's a representation on a real Hilbert space preserving the real inner product (so it means "orthogonal representation"). 
Apr 22 
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Can states on commutative Banach algebras be understood as probability measures?
In the usual correspondence for C*algebras you get moreover that the elements of $A$ correspond to random variables and evaluating a state on an element of $A$ corresponds to taking its expectation. But a nilpotent element of a commutative Banach algebra can't faithfully correspond to a random variable on any measurable space in a way compatible with multiplication. 
Apr 22 
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Can states on commutative Banach algebras be understood as probability measures?
What properties do you want the correspondence to have? I imagine if you ask too much there are difficulties with examples like $A = \mathbb{C}[x]/x^2$ (equipped with some norm, it doesn't really matter which). 
Apr 22 
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Monoidal Forgetful/Free Adjunction for $E_2$algebras
I'm confused. Suppose we work with just rings instead of ring spectra, so let $A, B$ be two commutative rings and let $f : A \to B$ be a morphism between them. Then $F$ is monoidal, but $U$ isn't. It's easiest to see this when $f$ is a field extension; $F$ preserves dimensions but $U$ multiplies them by the dimension of the extension. 
Apr 21 
awarded  Nice Answer 
Apr 20 
awarded  Nice Answer 
Apr 17 
awarded  Notable Question 
Apr 13 
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Maps to the group completion
Surely being an Hspace isn't a sufficient condition to admit a delooping? 
Apr 13 
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klinear abelian categories which are not categories of modules
On the other hand, that does suggest how to adapt the above construction to $B'$. We take the subalgebra of $B'$ consisting of sequences $a_i \in M_i(\mathbb{F}_2)$ such that $a_i$ is eventually a scalar and $\lim_{i \to \infty} a_i = a_1$. That might work. But it can be ruled out by removing $M_1(\mathbb{F}_2)$ from the product... and to rule out similarlooking constructions let's maybe only take the product over prime $i$. 
Apr 13 
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klinear abelian categories which are not categories of modules
@Eric: that does not work for $B$. The subalgebra $A$ you suggest is the subalgebra of sequences $a_i \in \mathbb{F}_2$ such that $\lim_{i \to \infty} a_i$ exists, and as such it has an extra map $A \to \mathbb{F}_2$ not factoring through the distinguished finite quotients of $B$, namely $\lim_{i \to \infty} a_i$ itself. The induced map $A \to B$ of Boolean rings corresponds topologically to the natural map from the StoneCech compactification of $\mathbb{N}$ to its onepoint compactification. 
Apr 13 
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klinear abelian categories which are not categories of modules
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Apr 13 
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klinear abelian categories which are not categories of modules
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Apr 13 
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klinear abelian categories which are not categories of modules
deleted 12 characters in body 
Apr 13 
answered  klinear abelian categories which are not categories of modules 
Apr 13 
awarded  Announcer 
Apr 13 
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klinear abelian categories which are not categories of modules
Also, in the question, by "equivalent" do you just mean as $k$linear categories or do you mean as $k$linear categories together with a choice of fiber functor? 