Seminar
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| Location: | SLMath: Eisenbud Auditorium |
Okounkov's conjecture via BPS Lie algebras
Given an arbitrary finite quiver Q, Maulik and Okounkov defined a new Yangian-style quantum group. It is built from the FRT formalism and their construction of R matrices on the cohomology of Nakajima quiver varieties, via the stable envelopes that Maulik and Okounkov also defined. Just as in the case of ordinary Yangians, there is a Lie algebra g_Q inside their new algebra, and the Yangian is a deformation of the current algebra of this Lie algebra. Outside of extended ADE type, numerous basic features of g_Q have remained mysterious since the outset of the subject, for example, the
dimensions of the graded pieces. A conjecture of Okounkov predicts that these dimensions are given by the coefficients of Kac's polynomials, which count isomorphism classes of absolutely indecomposable Q-representations over finite fields. I will explain a recent result, with Botta, stating that the Maulik-Okounkov Lie algebra associated to a quiver Q is isomorphic to the BPS Lie algebra associated to the tripled quiver of Q, along with its canonical cubic potential. Thanks to recent joint work of myself with Hennecart and Schlegel Mejia, this result implies Okounkov's conjecture, as well as determining the generators and relations of g_Q in terms of intersection cohomology of singular quiver varieties.