User sasha kirillov - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:09:50Z http://mathoverflow.net/feeds/user/10745 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40337/ingenuity-in-mathematics/47857#47857 Answer by Sasha Kirillov for Ingenuity in mathematics Sasha Kirillov 2010-12-01T03:38:56Z 2010-12-01T03:38:56Z <p>I like very much the proof of fundamental theorem of algebra (using the winding number around the origin), but it will probably take too long to explain... </p> http://mathoverflow.net/questions/46450/is-there-a-hope-for-formula-for-a-colored-homflypt-invariant-in-terms-of-homfly/46472#46472 Answer by Sasha Kirillov for Is there a (hope for) formula for a colored HOMFLYPT invariant in terms of HOMFLYPT invariants colored by the fundamental represenations? Sasha Kirillov 2010-11-18T11:35:49Z 2010-11-18T11:35:49Z <p>It depends on what you call "independent of knot combinatorics". Obviously one can get the invariant associated with a representation $V_\lambda\subset V^{\otimes k}$, where $V$ is the fundamental representation, by replacing every strand of the link by $k$ strands (cabling) colored by $V$ and then inserting the projector onto $V_\lambda$; since this projector can be written in terms of Hecke algebra operators (this is the $q$-version of Young symmetrizer), it can be presented graphically as a certain linear combination of $k-k$ tangles. This is what is done in the paper cited by Scott. Not sure if this is the answer you are looking for.</p> http://mathoverflow.net/questions/45881/whats-dual-torus-and-mirror-manifold/45895#45895 Answer by Sasha Kirillov for What's dual torus and mirror manifold? Sasha Kirillov 2010-11-13T04:48:56Z 2010-11-13T04:48:56Z <p>The usual answer is that the dual lattice is <code>$\check{\Gamma}=\{f\in V^* | f(\gamma)\in \mathbb{Z}\ \forall \gamma \in \Gamma\}$</code>. It is defined for any lattice $\Gamma\subset V$ - no extra information needed. </p> http://mathoverflow.net/questions/45708/discrete-g-connections/45712#45712 Answer by Sasha Kirillov for Discrete G-connections Sasha Kirillov 2010-11-11T15:31:30Z 2010-11-11T15:31:30Z <p>I'd begin by looking at Oeckl's "Discrete gauge theory": <a href="http://www.amazon.com/Discrete-Gauge-Theory-Lattices-Tqft/dp/1860945791" rel="nofollow">http://www.amazon.com/Discrete-Gauge-Theory-Lattices-Tqft/dp/1860945791</a></p> <p>For a finite group $G$, the notion of a $G$-connection is easy to define; it is usually done when you have not just a graph but a 2-complex and add a condition that the monodromy of connection around the boundary of each 2-cell is trivial. If you apply it to a cell decomposition of a closed surface, you get the space of gauge equivalence classes of $G$-bundles on the surface, so teh result does not depend on the choice of cell decomposition. This is a special (and well-known) example of Turaev-Viro theory, which in this case is also known as Chern-Simons theory with a finite gauge group (see paper of Freed and Quinn in CMP vol. 156).</p> http://mathoverflow.net/questions/19930/writing-papers-in-pre-latex-era/47834#47834 Comment by Sasha Kirillov Sasha Kirillov 2010-12-01T03:52:41Z 2010-12-01T03:52:41Z Oh... what memories. I used it too at some point. http://mathoverflow.net/questions/47690/are-the-extra-vertices-in-nakajimas-doubling-of-a-quiver-related-to-langlands-du Comment by Sasha Kirillov Sasha Kirillov 2010-12-01T03:22:06Z 2010-12-01T03:22:06Z A correction: $M(v,w)$ is not exactly the moduli space of reps of $Q^\heartsuit$. Namely, its points are representations of $Q^\heartsuit$ but the notion of isomorphism is different: points correspond to orbits of $GL(v)$ action, not $GL(v)\times GL(w)$ action. http://mathoverflow.net/questions/33681/is-there-a-quantum-hermite-reciprocity/33708#33708 Comment by Sasha Kirillov Sasha Kirillov 2010-11-18T11:55:09Z 2010-11-18T11:55:09Z One can define $Sym^n(V_\lambda)$ as the subspace <code>$\{v: R&#95;iv=c&#95;\lambda v\}$</code>, where $c_\lambda$ is the (easily computable) eigenvalue of $R$ on $v_\lambda\otimes v_\lambda\in V_\lambda\otimes V_\lambda$. The difficulty of generalizing it to reducible representations is that $c_\lambda$ depends on $\lambda$.... http://mathoverflow.net/questions/45708/discrete-g-connections/45712#45712 Comment by Sasha Kirillov Sasha Kirillov 2010-11-13T04:30:34Z 2010-11-13T04:30:34Z For a finite group, every connection in a $G$-bundle over a manifold is automatically flat.