Seminar in Representation Theory
P/Q-key polynomials
2026/01/20 (Tue.) 15:00~16:00
- Date : 2026/01/20 (Tue.) 15:00~16:00
- Location : Seminar Room 638, Institute of Mathematics (NTU Campus)
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Speaker : Travis Scrimshaw (Hokkaido University)
- Organizer : Chun-Ju Lai (AS)
- Abstract : The Borel-Weil(-Bott) theorem for the general linear group G (over the complex numbers) gives an important geometric construction of a highest weight representation V of G by using a line bundle over the flag variety G/B, where B is the subgroup of upper triangular matrices, whose fiber is a one dimensional B-representation. By restricting to a Schubert variety, the closure of a B-orbit of G/B and indexed by permutations, we obtain Demazure modules that give a filtration of V and whose characters are called key polynomials. By using the geometry, we can give a recursive description for these polynomials using divided difference operators, and Kashiwara’s theory of crystal bases gives a combinatorial description of the representation theory. In this talk, we will give an analogous algebraic and crystal-theoretic construction that is related with the Lie superalgebra of type Q using the orbit closures of the orthogonal and symplectic groups in (the cohomology of) G/B. We call these polynomials P-key and Q-key polynomials. We will discuss a number of open problems and conjectures about these polynomials and extensions to the K-theory. This is based on joint work with Eric Marberg.
