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Session S10 - Categorification, Higher Representation Theory, and Homological Knot Invariants

Friday, July 16, 18:35 ~ 19:10 UTC-3

Combinatorial invariance conjecture for $\widetilde{A}_2$

David Plaza

Universidad de Talca, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakbd1ed0cbdb296c00b5d2024c2f2e505e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addybd1ed0cbdb296c00b5d2024c2f2e505e = 'd&#97;v&#105;dr&#105;c&#97;rd&#111;pl&#97;z&#97;' + '&#64;'; addybd1ed0cbdb296c00b5d2024c2f2e505e = addybd1ed0cbdb296c00b5d2024c2f2e505e + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textbd1ed0cbdb296c00b5d2024c2f2e505e = 'd&#97;v&#105;dr&#105;c&#97;rd&#111;pl&#97;z&#97;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakbd1ed0cbdb296c00b5d2024c2f2e505e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addybd1ed0cbdb296c00b5d2024c2f2e505e + '\'>'+addy_textbd1ed0cbdb296c00b5d2024c2f2e505e+'<\/a>';

The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer) predicts that if $[x,y]$ and $[x',y']$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan-Lusztig polynomials are equal, i.e., $P_{x,y}(q)=P_{x',y'}(q)$. In this talk we prove this conjecture for the affine Weyl group of type $\widetilde{A}_2$. This is the first non-trivial case where this conjecture is verified for an infinite group.

Joint work with Gastón Burrul (The University of Sidney, Australia), and Nicolás Libedinsky (Universidad de Chile, Chile).