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/Logic/Number Theory seminar series.

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