can the Newton's identities and Dodgson's condensations be proved by Gessel-Viennot's lemma? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:49:13Z http://mathoverflow.net/feeds/question/45145 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45145/can-the-newtons-identities-and-dodgsons-condensations-be-proved-by-gessel-vienn can the Newton's identities and Dodgson's condensations be proved by Gessel-Viennot's lemma? zhaoliang 2010-11-07T07:20:52Z 2010-11-07T17:04:59Z <p>Gessel-Viennot's simple but powerful lemma has many striking applications, such as counting noninsecting paths , proving the Jacob-Trudi's identities, and solving the aztec diamond problem. So I wonder wether it can also be used to prove the Dodgson's condensation and Newton's indentities. Also, if you know other theorems or identities that can be solved by this lemma, please let me konw...</p> http://mathoverflow.net/questions/45145/can-the-newtons-identities-and-dodgsons-condensations-be-proved-by-gessel-vienn/45148#45148 Answer by Leonid Petrov for can the Newton's identities and Dodgson's condensations be proved by Gessel-Viennot's lemma? Leonid Petrov 2010-11-07T08:23:10Z 2010-11-07T08:23:10Z <p>A little off-topic: Gessel-Viennot's lemma was also discovered by Karlin and McGregor (1959, "Coincidence Probabilities") and was used to construct dynamics of noncolliding systems of particles: take N identical particles evolving independently (e.g., under a one-dimensional diffusion) and impose the condition that their trajectories do not intersect. Under some conditions you will get new interesting Markov particle dynamics with determinantal formulas for the semigroup and for the dynamical correlation functions, see also Koenig arXiv:math/0403090.</p> <p>Concerning the original question: Gessel-Viennot's lemma provides a powerful combinatorial formalism to produce totally positive matrices, and in the case of Toeplitz matrices any such totally positive matrix can be obtained this way. So this could possibly lead to the proof of the facts that you need.</p> http://mathoverflow.net/questions/45145/can-the-newtons-identities-and-dodgsons-condensations-be-proved-by-gessel-vienn/45158#45158 Answer by Martin Rubey for can the Newton's identities and Dodgson's condensations be proved by Gessel-Viennot's lemma? Martin Rubey 2010-11-07T11:19:47Z 2010-11-07T11:19:47Z <p>I think that <a href="http://arxiv.org/abs/1010.3860" rel="nofollow">Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively</a> by Markus Fulmek is your friend.</p>