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May
3
answered How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?
May
1
comment Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms
Insofar as these are 1-complexes not 2-complexes, are these containment questions well-defined? (They may be, in which case I'd like to hear the well definition.)
May
1
comment For $\mathfrak g$ A Lie algebra of type $ E_7 $, $\mathfrak h $ a Cartan subalgebra and $\Delta$ the resulting root system, does $ Aut(\mathfrak g,\mathfrak h)\rightarrow Aut(\Delta) $ split over the Weyl group?
math.cornell.edu/~allenk/courses/16spring/Construction11.pdf
Apr
30
comment Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)
First prove, using the map $G/T \to G/N(T)$, that the space $G/N(T)$ has only even cohomology but $\chi=1$ hence trivial rational cohomology. Then use Leray-Hirsch on $EG/N(T) \to EG/G$ to show they have the same rational cohomology. Finally, use the Galois covering space $EG/T \to EG/N(T)$ to show $EG/N(T)$'s cohomology is the $W$-invariants in that of $EG/T$.
Apr
30
answered Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)
Apr
27
comment Does the support of a regular holonomic D-module always have finitely many orbits?
D-modules make sense on spaces that have no group action, so this question doesn't make much sense. Take, for example, the $\mathcal D$-module $\mathcal O_{\mathbb A^1}$, and use the trivial group action on ${\mathbb A^1}$.
Apr
26
comment Rank diagrams of permutations $w \in S_{m}$ in the study of complete flag varieties
Instead of "some pair of flags", use the $GL(n)$-action to turn one flag into the base flag, the other into $w\cdot$ the base flag for some $w\in S_n$. That this is achievable (and $w$ is unique) is the Bruhat decomposition.
Apr
26
comment Equivariant and orbifold Chern classes
Also, your "canonical choice" only works if your orbifold is generically a manifold, not e.g. $pt/\Gamma$.
Apr
26
comment Equivariant and orbifold Chern classes
Say we declare Q4 to be an isomorphism, and thereby give an answer to Q1. Then Q2 and Q3 are the same question, and the answer is yes, there exist equivariant characteristic classes (they're the ordinary characteristic classes of Borel-construction(the bundle map)).
Apr
24
comment Rank diagrams of permutations $w \in S_{m}$ in the study of complete flag varieties
For the first, you're going to need to state a definition of "rank diagram". For the second, you presumably want to say "$dim()$" equals something. In any case, any results of this sort are going to follow from the Bruhat decomposition, see e.g. Fulton's "Young Tableaux".
Apr
23
revised Cotangent complex of certain dg-scheme
added 1 character in body
Apr
22
comment Order of metaplectic operator
The order may now be $2n$.
Apr
16
comment Grothendieck class and Normalization
Maybe you should consider seminormalizing first. I expect that shouldn't change the Grothendieck class at all.
Apr
16
revised Checking smoothness of the components of a highly symmetric scheme via quotient?
added 383 characters in body
Apr
16
answered Checking smoothness of the components of a highly symmetric scheme via quotient?
Apr
16
comment Passing from T-equivariant to G-equivariant cohomology
Two things. The relevant map of spaces here is $X_T := (X \times EG)/T \to X_G := (X\times EG)/G$ with fiber $G/T$ or homotopically $G/B$. So $H^*(X_T)$ can be computed from a spectral sequence from $H^*(X_G)$. If for some reason you knew already the latter had even-degree cohomology... The other thing is that your map factors through $H^*_{N(T)}(X;\mathbb Z)$, and I suspect any kernel has to happen already there.
Apr
16
comment Grothendieck class and Normalization
If $X$ is a nodal cubic curve and $\tilde X$ is its normalization, then $[\tilde X] = [X] + [pt]$. Is that "behaving well"?
Apr
10
comment Projections of orbifolds
Actually, the story starts before (and influenced) [Kostant 1970] with the Schur-Horn theorem from the '30s and '40s, which is the $G=SU(n)$ case the OP focused on.
Apr
6
comment Is there a name for this construction from two representations?
Indeed, is your $G$ abelian? If not I'm pretty sure the closest thing is the vector space (not $G$-rep) $V \otimes_{k[G]} W$. Or maybe you'd prefer $Hom_G(V^*,W)$ (assuming $V$ finite-dimensional). Either way you have to convert a left $G$-rep to a right $G$-rep, but of course there's a natural way to do that. (And if $G$ is abelian, a second such.)
Apr
5
comment If the two smallest eigenvalues of the Laplacian matrix of a network are equal to zero, then does it mean that the network is not connected?
That it's disconnected, yes. This is standard in treatments of the subject.