Discrete Geometry and Combinatorics Seminar Spring 2019

All seminars (unless otherwise stated) will take place on Tuesdays at 5pm in Maths Room 707 (25 Gordon Street) - see the map for further details. If you require any more information on the Discrete Geometry and Combinatorics seminars, please contact Dr John Talbot (e-mail: j.talbot AT ucl.ac.uk or tel: 020-7679-4102).

05 February 2019

Speaker: Francois Bergeron (UQAM)

Title: Why is Schur positivity interesting?

Not only do symmetric polynomials  play a fundamental role in many domains of mathematics (Galois Theory, Representation Theory, Algebraic Geometry, Algebraic Combinatorics, etc.), but they also occur naturally in several areas of Theoretical Physics (Quantum Mechanics, Quantum Many-Body Systems, Particle Physics, etc.). Under various specializations, identities in the algebra of symmetric polynomials are an especially rich source of combinatorial identities; and Schur polynomials are of particular interest in these identities. Indeed, they constitute the "most interesting linear basis" of the algebra of  symmetric polynomials. One says that a symmetric is "Schur positive" if it expands with positive coefficients (most often integers) in the Schur polynomial basis. Exploiting many examples of interesting contexts in which symmetric polynomials play a natural role, we will illustrate why Schur positivity is a phenomenon that is both important and of frequent interest. On the other hand, we will show that it is also a very rare phenomenon in the absence of some extra hypothesis (such as that they arise from representation theory). This talk is intended to be accessible to a general audience.