Kazhdan-Lusztig Positivity Conjectures - The Algebraic Viewpoint

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• Robert Spencer, University of Cambridge
• Friday 14 February 2020, 15:00-16:00
• CMS, MR13.

If you have a question about this talk, please contact Liam Jolliffe.

Kazhdan-Lusztig polynomials are easy to compute elements of Z[v] giving the coefficients of the “Kazhdan-Lusztig basis” of the Hecke algebra in terms of the standard basis. A famous conjecture called the Kazhdan-Lusztig Positivity Conjecture states that the coefficients of the polynomials are non-negative. This was proven in the 1980s using D-modules, perverse sheaves and other geometrical constructions. Much later in the 2010s, a more algebraic proof based off Soergel bi-modules and Hodge theory emerged. In this introductory talk we will define the Kazhdan-Lusztig polynomials from the ground up, give an overview of the conjectures (some true, some false) surrounding them, define the basics of Soergel bi-module theory and give a very light sketch of the algebraic proof of the Positivity Conjecture (and a couple of others). Time permitting, we will mention some of the offshoots from this theory of current interest.

This talk is part of the Junior Algebra and Number Theory seminar series.

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