Richard P. Stanley Seminar in Combinatorics: Rational Catalan Numbers for Complex Reflection Groups

SEMINARS, HARVARD-MIT COMBINATORICS

Speaker:

Weston Miller - UT Dallas

The spetsial complex reflection groups are complex reflection groups that behave as if they were the Weyl group for some connected reductive algebraic group. Analogs of unipotent characters and Lusztig’s Fourier transform can be defined combinatorially for these groups, allowing some techniques from the representation theory of finite groups of Lie type to be extended to spetsial complex reflection groups.

In a recent paper, Galashin, Lam, Trinh, and Williams introduced a family of rational noncrossing objects for finite Coxeter groups. The proof that these objects are counted by rational Coxeter-Catalan numbers used Hecke algebra traces to compute the point count of braid Richardson varieties. Assuming standard conjectures, I prove that this trace technique extends to irreducible spetsial complex reflection groups. That is, I show that the trace of a power of a Coxeter element still produces a rational Catalan number. I’ll also discuss the related rational parking problem.

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For more info, see https://math.mit.edu/combin/