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bio website math.berkeley.edu/~qchu
location Berkeley, CA
age 24
visits member for 5 years, 6 months
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I am a third-year graduate student at UC Berkeley. I'm interested in the interplay between homotopy theory, quantum field theory, manifold topology, and higher category theory.


1d
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment Maps to the group completion
Surely being an H-space isn't a sufficient condition to admit a delooping?
Apr
13
comment k-linear 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 similar-looking constructions let's maybe only take the product over prime $i$.
Apr
13
comment k-linear 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 Stone-Cech compactification of $\mathbb{N}$ to its one-point compactification.
Apr
13
revised k-linear abelian categories which are not categories of modules
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Apr
13
revised k-linear abelian categories which are not categories of modules
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Apr
13
revised k-linear abelian categories which are not categories of modules
deleted 12 characters in body
Apr
13
answered k-linear abelian categories which are not categories of modules
Apr
13
awarded  Announcer
Apr
13
comment k-linear 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?