Abstract:
Let \$u\$ and \$v\$ be two permutations of the numbers \$1,\ldots,n\$. Associated to \$u\$ and \$v\$ is a polynomial \$P_{uv}\$, called the Kazhdan-Lusztig polynomial, which encodes numerical invariants that are central in geometric representation theory. The coefficients of \$P_{uv}\$ simultaneously describe the singularities of Schubert varieties, the structure of Hecke algebras, and the representation theory of Lie algebras. Associated to \$u\$ and \$v\$ is another object, the Bruhat graph of \$(u,v)\$, which is a directed graph describing the transpositions taking \$u\$ to \$v\$.
The combinatorial invariance conjecture (CIC) of Dyer and Lusztig asserts that the Bruhat graph of \$(u,v)\$ uniquely determines \$P_{uv}\$. Recently, Geordie Williamson and Google DeepMind applied machine learning techniques to this problem. Using those techniques, they conjectured an explicit recursion that would compute \$P_{uv}\$ from the Bruhat graph and thereby prove the CIC. In joint work with Christian Gaetz, we prove the Williamson-DeepMind conjecture in the case where \$u\$ is the identity permutation. Along the way, we prove two new identities for the Kazhdan--Lusztig \$R\$ polynomials, one of which implies new cases of the CIC.
Note: The main talk will begin at 3:30. There is no pre-seminar.
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Meeting ID: 915 4733 5974