13,363 reputation
12983
bio website front.math.ucdavis.edu/…
location Cornell
age 44
visits member for 4 years, 9 months
seen 4 hours ago
Math professor at Cornell, PhD 1996

18h
comment Theorems for nothing (and the proofs for free)
Finite division rings commute.
Jul
17
comment GIT over integers
For the first one, I definitely get confused in the situation that $G$ doesn't have enough $\mathbb Z$-points. Say $\mathbb G_m$ acts on $Spec\ \mathbb Z[x,y]$ with weights $+1,-1$ on the coordinates. What does $A^G$ mean here, where the $\mathbb Z$-points of $\mathbb G_m$ are just $\pm 1$? Do we first think of the ring $A$ as an algebraic variety of infinite type, on which $G$ is acting algebraically?
Jul
16
comment Is every Noetherian *connected* ring a quotient of a Noetherian domain?
Could you embed the primary components disjointly in a common space, then start gluing that space to itself?
Jul
16
comment Rational normal curves on Grassmanians
I'm confused: I would have thought that [BKT] would give a birational isomorphism of this space with $G(2k,n)$, which obviously isn't happening here.
Jul
15
comment Sum over growing Young tableaux
If I understand right, for every infinite sequence $\lambda$, you have an associated $f$? Anyway all that occurs to me is the hooklength formula, which says that $f_{\lambda_i}/f_{\lambda_{i-1}}$ is a reasonably simple fraction.
Jul
14
comment Vector bundles, Higgs bundles and the Langlands program
Tiny thing: in general, a $\mathcal D$-module on $M$ has a characteristic cycle on $T^* M$.
Jul
14
answered Semicontinuity of degree of fibers for a proper map
Jul
14
comment counting the number of ordered pairs in a permutohedron
I don't know that this question has a particularly satisfying answer, but I do know that the order you're using is opposite "weak Bruhat order", and recommend you reverse it before confusing people. Have you computed the first few numbers and checked the OEIS?
Jul
13
comment Blow-up and tangent cone
I agree with Jason. I would first look at $N_{E/X} \to N_{p/Y} \hookrightarrow T_p Y$, and observe that this is dilation-equivariant. I guess your question is whether this can actually be projectivized? How about the map $\mathbb A^1 \to \mathbb A^1$, $z\mapsto z^3$, $p = 0$?
Jul
9
comment Surprising connection between linear algebra and graph theory
Linear algebra is so powerful that we use it to study all other mathematical objects (de Rham cohomology comes to mind), graphs just being one example.
Jul
8
comment Cohomology of Bott-Samelson varieties?
This is exactly what the original Bott and Samelson paper is about, and you should look at it. (Only later did people even realize that these manifolds are algebraic varieties, much less that they provide resolutions of singularities.)
Jul
2
awarded  Inquisitive
Jul
2
awarded  Curious
Jun
22
comment maximal chain in (strong) Bruhat order satisfying constraint
People talk about "left weak order" and "right weak order". Strong order is symmetric under taking inverses, but the weak orders aren't.
Jun
19
revised Positroids and Totally Nonnegative Complex Grassmanian
disambiguated last sentence
Jun
17
answered Which linear combinations of simple roots are roots
Jun
14
comment A property of compact involutions of semi-simple Lie algebras?
It seems like everything you're asking transforms nicely under the Weyl group, so one could replace "highest root" by "any long root", even a simple one.
Jun
10
revised Euler characteristics with and without compact support of algebraic varieties
stratUM
Jun
8
comment R-linear representations of sl(2,C)
Oops: my question was essentially the answer for $\mathfrak{sl}_2(\mathbb R)$, not its square $\mathfrak{sl}_2(\mathbb C)$. I should have been tipped off by your mention of Lorentz.
Jun
8
comment R-linear representations of sl(2,C)
I don't know about a reference, but here goes. A complex irrep is the complexification of a real irrep iff it's self-dual, and then, if the unique invariant bilinear form is symmetric. (The others are quaternionic, and in particular even-dimensional.) On dominant weights, duality is given by $-w_0$, so for $sl_2$ they're all self-dual. The odd-dimensional ones $V_{2n}$ are obviously real. The 2-d one $V_1$ is obviously quaternionic. So each $V_{2n+1} \leq V_{2n} \otimes V_1$ is quaternionic. Summary: each odd-dimensional irrep is the complexification of a real irrep, and that's all of them.