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13088
bio website front.math.ucdavis.edu/…
location Cornell
age 45
visits member for 5 years
seen 17 mins ago
Math professor at Cornell, PhD 1996

Oct
21
comment scheme of commuting matrices
For $r=2$ it's irreducible (Richardson's theorem, as I recall). For $r=3$ it's terrible even for fairly small $n$, because of the close connection with the Hilbert scheme of $n$ points in $\mathbb A^r$.
Oct
19
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
The cluster story continues from there to combinatorics in front.math.ucdavis.edu/1108.1776 "Subword complexes, cluster complexes, and generalized multi-associahedra" and then to geometry in front.math.ucdavis.edu/1404.4671 "Brick manifolds and toric varieties of brick polytopes".
Oct
15
awarded  Yearling
Oct
13
comment The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$
Their comments were tied, so I upvoted yours to make it a 3-way tie.
Oct
11
comment Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
The "simple iff antidominant" statement is only in the presence of the $\lambda$ integral assumption, right?
Oct
9
comment When is the Hodge diamond concentrated in $H^{n,n}$'s?
As others say, it's very natural to ask that all cohomology be in $H^{p,p}$, and therefore that the odd cohomology vanishes; I think it very strange to ask this only of the even cohomology.
Oct
9
comment No basis change in a fusion ring allowed?
There are lots of other rings out there, e.g. $H^*(G/P)$, that are pretty boring as rings but interesting as rings-with-bases.
Oct
6
comment Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
Exists a formula, or should exist such a formula? Do you think a reference exists? (This is really what motivates my original question.)
Oct
5
comment G-equivariant coherent sheaves on Bott-Samelson Resolutions
Yes, this is only $B$-equivariant, and yes, $Rf_*\mathcal O = \mathcal O$, even when the word is not reduced. One place to read about such things is Brion and Kumar's excellent book (on Frobenius splitting).
Oct
5
comment Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
(1) When $G\geq H$, a $(\mathfrak{g},H)$-module $V$ is one with actions of $\mathfrak{g}$ and of $H$, from each of which one can derive an action of $\mathfrak h$; the two actions should agree. (2) As for what extra information it should contain beyond that of a $\mathfrak{g}$-representation: a $(\mathfrak g,B)$-irrep is in category $\mathcal O$, unless I'm very confused. (3) My weight $\lambda$ was assumed integral, so better to think of it as a weight for $T$ (or $B$ since $B/[B,B] \equiv T$) than for $\mathfrak t$.
Oct
4
awarded  Nice Question
Oct
4
comment Localization on orbit type submanifolds (generalization of Atiyah-Bott-Berline-Vergne)
I think the closest is Paul-Emile Paradan's take on Witten's nonabelian localization, at the critical points of $|\Phi|^2$, where $\Phi$ is the $G$-moment map ($M$ is symplectic). If $G$ is abelian, $M$ compact, and we add a large constant value to $\Phi$, this reduces to A-B/B-V.
Oct
4
revised Derivation of Blattner's conjecture in the Beilinson-Bernstein picture
added 103 characters in body
Oct
4
comment Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
That "trivial" case is exactly when $\lambda$ (which I assumed integral) is regular antidominant, right? At least without your "generalized".
Oct
4
revised Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
added 10 characters in body
Oct
4
comment Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
I meant to have reductive there; I'll put it in. Yes your $L(\lambda)$ is my $V_\lambda$, insofar as they're unique. What could there be to say about "the nature of the $B$-action"? 1-d reps are of $B/B' \cong T$, so weights, like $\lambda$.
Oct
3
asked Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
Sep
30
awarded  Explainer
Sep
29
revised Derivation of Blattner's conjecture in the Beilinson-Bernstein picture
added 389 characters in body
Sep
29
comment Derivation of Blattner's conjecture in the Beilinson-Bernstein picture
She references the Blattner formula, but doesn't derive it, as far as I could tell.