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336109
bio website mit.edu/~darij/www
location Karlsruhe (home), Munich (until summer 2011), Cambridge/MA (2011-)
age 26
visits member for 5 years, 3 months
seen 4 hours ago
I'm just here for asking stupid questions.

2d
revised Why does this antisymmetric product factor out a determinant?
added 3 characters in body
Mar
14
revised Rees algebra isomorphism
added 1 character in body
Mar
14
comment Rees algebra isomorphism
... isomorphic. Dropping the $t^k$, this means proving that $I^k / \left(J \cap I^k\right) \cong \overline{I}^k$. But the latter is clear, since $\overline{I}^k = \overline{I^k}$ (the projection of $I^k$ onto $\overline{R}$). Once you have these isomorphisms for all $k$, you can take their direct sum and see that it is an algebra isomorphism (easy, because they are very explicit). So the answer to the first question is "Yes". Similarly for the second question.
Mar
14
comment Rees algebra isomorphism
Correct me if I am wrong, but I believe there are obvious isomorphisms obtained by just considering the different algebras degree by degree (since everything is graded). For instance, for $k \in \mathbb{N}$, the $k$-th homogeneous component of $R\left[It\right] / \left(JR\left[t\right] \cap R\left[It\right]\right)$ is $I^k t^k / \left(Jt^k \cap I^k t^k\right)$, while the $k$-th homogeneous component of $\overline{R}/\overline{I}$ (where $\overline{R} = R/J$, and where $\overline{I}$ is the projection of $I$ onto $\overline{R}$) is $\overline{I}^k t^k$. It remains to show that these are ...
Mar
14
comment Dynamics of RSK
Oh! So this wasn't a case of "turning around an octahedron" after all? On the upside, that question perfectly fits the upcoming conference.
Mar
12
comment Dynamics of RSK
Notice that the composition $\operatorname{revrectsum} \circ \operatorname{diagsum}$ has a simple description: It is the inverse of the map sending a matrix $\left(x_{i,j}\right)_{i,j}$ to $\left(\sum\limits_{k=1}^{i-1} x_{k,j} + \sum\limits_{k=1}^{j-1} x_{i,k} + x_{i,j}\right)_{i,j}$ (now I am back to using your tropical notations). So it can be called "co-hook sum".
Mar
12
comment Dynamics of RSK
OK, it does not look like double evacuation, because it does not act on the two halves of the matrix separately (at least not on the right two halves). But generally it appears to me that $\operatorname{RSK} \circ \operatorname{revrectsum} \circ \operatorname{diagsum}$ is the simpler map. For example, it maps $\left(\begin{matrix} x & y \\ z & w \end{matrix}\right)$ to $\left(\begin{matrix} \frac{y+z}{x} & y \\ z & \frac{w}{y+z} \end{matrix}\right)$ (in the birational setting; to go tropical, replace $+$ by $\min$ and $/$ by $-$).
Mar
12
comment Dynamics of RSK
Very nice! Do you have an idea if this (or, rather, the conjugate version $\operatorname{RSK} \circ \operatorname{revrectsum} \circ \operatorname{diagsum}$) could be a (double) evacuation map in disguise?
Mar
12
comment What's the analogue of a Young symmetrizer in the Brauer algebra?
A few years ago I have asked Arun Ram a similar question (I was looking for a lift to the partition algebra, not the Brauer algebra), and he referred me to one of his oldest papers (this is what he said). It might be Arun Ram and Hans Wenzl, Matrix Units for Centralizer Algebras ( sciencedirect.com/science/article/pii/002186939290109Y ). I did not, unfortunately, dig into this; I am not sure if the paper actually settles the question (it seems to be more about generalizing seminormal units).
Mar
11
comment Fixed space of the square of a symmetric matrix over $\mathbb{F}_2$
What if $M$ is alternating?
Mar
11
awarded  linear-algebra
Mar
11
answered Traces and projectors
Mar
11
comment Traces and projectors
$V$ is $2$-dimensional with basis $\left(e,f\right)$. The projectors are projecting on $\mathbb C e$ and $\mathbb C f$, each with kernel $e+f$. The map $T$ projects on $\mathbb C \left(e+f\right)$. This is a counterexample, right?
Mar
9
reviewed No Action Needed What is $K_2(\mathbb{Z}[x,x^{-1}])$?
Mar
4
comment Communal problem books
@MoritzFirsching: These are "Books of abstracts" and "Guest books" in English. None of them seems to be a place for exchanging problems.
Mar
4
comment Communal problem books
@MoritzFirsching: Where exactly? I cannot find them in there (though they might just be tagged under the wrong tag).
Mar
3
comment Communal problem books
Oberwolfach still has its book. I don't know if (parts of) it has ever been published, though I suspect most successful solutions have.
Mar
3
comment Publication in proceedings
@AlexDegtyarev: Because proceedings volumes are usually hard to get (libraries don't always have them, and publishers don't normally put them on the web even behind paywalls), and most of the time, people come to conferences with talks which announce a result that is to be polished and published later. That is different in computer science, but I was talking about mathematics here.
Mar
3
comment Publication in proceedings
When I see a proceedings paper, I check if there is a journal paper elaborating on it. If not, I start wondering whether it has been disproven before it could get published...
Feb
28
comment Flooding a cycle digraph via chip-firing: $n^{k-1} + n^{k-2} + \cdots + 1$ bound (a Norway 1998-99 problem generalized)
This is an even better proof. Thanks a lot!