11,280 reputation
233103
bio website mit.edu/~darij/www
location Karlsruhe (home), Munich (until summer 2011), Cambridge/MA (2011-)
age 25
visits member for 4 years, 10 months
seen 10 hours ago
I'm just here for asking stupid questions.

2d
comment Obscure Names in Mathematics
Speaking of tropical geometry: the Aztec diamond and the arctic circle theorem ( en.wikipedia.org/wiki/Aztec_diamond ).
Oct
15
comment Isotypic components of the action of the symmetric group on polynomials
"it is a regular representation": why?
Oct
15
comment Isotypic components of the action of the symmetric group on polynomials
I think he wants a decomposition into $S_n$-submodules.
Oct
9
comment Generalized Cauchy-Binet sum over a fixed subset of indices
Isn't it just the Cauchy-Binet formula applied to the matrices $A_{[m],[n-j]}$ and $B_{[n-j],[m]}$ instead of $A$ and $B$ ?
Oct
9
comment Generalized Cauchy-Binet sum over a fixed subset of indices
I think the simplest answer will be $\det\left(A_{[m],[n-j]} B_{[n-j],[m]}\right)$.
Oct
8
comment Does this permanent have a closed form?
For $n = 2$, the result is a fraction whose numerator is an irreducible degree-4 polynomial. If anything factors for higher $n$, then I don't think the factors will be any simpler than this polynomials...
Oct
8
comment Deligne's exterior power
What do you want to do with two variables? If $\alpha x + \beta y = 0$, then $\alpha x \wedge \beta y = 0$.
Oct
8
comment Deligne's exterior power
What happens if you replace my $k$ by the polynomial ring $\mathbb{F}_2\left[\alpha,\beta,\gamma\right]$, change the definition of $L$ to $L = M / \left< \alpha x + \beta y + \gamma z, \alpha^2 x, \alpha^2 y, \alpha^2 z, \beta^2 x, \beta^2 y, \beta^2 z, \gamma^2 x, \gamma^2 y, \gamma^2 z \right>$, and change the image of $f$ to $k / \left(\alpha^2, \beta^2, \gamma^2\right)$ ?
Oct
8
comment Deligne's exterior power
Nice. I tried to simplify the calculation like that, but I failed, so I decided to keep the symmetric form.
Oct
8
comment Non-Convex Polygons with “Antipodal Visibility”
Take a triangle $ABC$ with $\left|AB\right| = \left|AC\right|$ and with $\measuredangle BAC$ very small (thus an isosceles triangle with an apex very high up). Let $H$ be its orthocenter. I am fairly sure $ABHC$ has antipodal visibility (and is not convex).
Oct
8
answered Deligne's exterior power
Oct
8
comment Is there a Gröbner basis analogue that exists for vector spaces?
Thank you; I meant "nonlinear" indeed.
Oct
7
revised Is there a Gröbner basis analogue that exists for vector spaces?
added 1 character in body; edited tags
Oct
7
comment Is there a Gröbner basis analogue that exists for vector spaces?
V is a subspace of the vector space on which your coordinate system is defined? Then a basis $\left(v_1, v_2, \ldots, v_n\right)$ has the property that you desire if and only if it is a permutation of the list of rows of a matrix in row-echelon form. I am fairly sure that this is well-known; many people motivate Groebner basis theory as a noncommutative analogue of classical linear algebra (Groebner basis <~~> row echelon form; reduced Groebner basis <~~> reduced row echelon form).
Oct
2
comment Is there a bijection of permutations onto mathematical objects that preserve information about descents?
The map that sends a permutation to its descent set itself is a very useful thing. It is the map $S_n \to Q_n$ in Loday/Ronco "Hopf Algebra of the Planar Binary Trees" ( sciencedirect.com/science/article/pii/S0001870898917595 ), and factors through $Y_n$ as shown on page 1 of that paper.
Sep
30
awarded  Explainer
Sep
30
awarded  Nice Question
Sep
30
revised Anti-arithmetic product of symmetric functions: (why) is it integral?
added 247 characters in body
Sep
29
asked Anti-arithmetic product of symmetric functions: (why) is it integral?
Sep
28
comment Which math paper maximizes the ratio (importance)/(length)?
@DanielLitt: Thank you, but I fear you overestimated my progress. I still don't know what a derived category is and don't feel that I have the time and peace of mind to read myself into them properly.